<?xml version="1.0" encoding="us-ascii"?><InstanceReport xmlns:xsd="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"><Version>2.4.0.8</Version><ReportLongName>113 - Disclosure - Legal Proceedings</ReportLongName><DisplayLabelColumn>true</DisplayLabelColumn><ShowElementNames>false</ShowElementNames><RoundingOption /><HasEmbeddedReports>false</HasEmbeddedReports><Columns><Column FlagID="0"><Id>1</Id><IsAbstractGroupTitle>false</IsAbstractGroupTitle><LabelSeparator>

</LabelSeparator><CurrencyCode /><FootnoteIndexer /><hasSegments>false</hasSegments><hasScenarios>false</hasScenarios><MCU><KeyName /><CurrencySymbol /><contextRef><ContextID>eol_PE10781---1310-K0007_STD_364_20130628_0</ContextID><EntitySchema>http://www.sec.gov/CIK</EntitySchema><EntityValue>0000106040</EntityValue><PeriodDisplayName /><PeriodType>duration</PeriodType><PeriodStartDate>2012-06-30T00:00:00</PeriodStartDate><PeriodEndDate>2013-06-28T00:00:00</PeriodEndDate><Segments /><Scenarios /></contextRef><UPS /><CurrencyCode /><OriginalCurrencyCode /></MCU><CurrencySymbol /><Labels><Label Key="CalendarSupplement" Id="0" Label="12 Months Ended" /><Label Key="Calendar" Id="1" Label="Jun. 28, 2013" /></Labels></Column></Columns><Rows><Row FlagID="0"><Id>1</Id><IsAbstractGroupTitle>true</IsAbstractGroupTitle><LabelSeparator>

</LabelSeparator><Level>1</Level><ElementName>us-gaap_CommitmentsAndContingenciesDisclosureAbstract</ElementName><ElementPrefix>us-gaap_</ElementPrefix><IsBaseElement>true</IsBaseElement><BalanceType>na</BalanceType><PeriodType>duration</PeriodType><IsReportTitle>false</IsReportTitle><IsSegmentTitle>false</IsSegmentTitle><IsCalendarTitle>false</IsCalendarTitle><IsEquityPrevioslyReportedAsRow>false</IsEquityPrevioslyReportedAsRow><IsEquityAdjustmentRow>false</IsEquityAdjustmentRow><IsBeginningBalance>false</IsBeginningBalance><IsEndingBalance>false</IsEndingBalance><IsReverseSign>false</IsReverseSign><FootnoteIndexer /><Cells><Cell FlagID="0" ContextID="" UnitID=""><Id>1</Id><IsNumeric>false</IsNumeric><IsRatio>false</IsRatio><DisplayZeroAsNone>false</DisplayZeroAsNone><NumericAmount>0</NumericAmount><RoundedNumericAmount>0</RoundedNumericAmount><NonNumbericText /><FootnoteIndexer /><CurrencyCode /><CurrencySymbol /><IsIndependantCurrency>false</IsIndependantCurrency><ShowCurrencySymbol>false</ShowCurrencySymbol><DisplayDateInUSFormat>false</DisplayDateInUSFormat></Cell></Cells><ElementDataType>xbrli:stringItemType</ElementDataType><SimpleDataType>string</SimpleDataType><IsTotalLabel>false</IsTotalLabel><UnitID>0</UnitID><Label>Commitments And Contingencies Disclosure [Abstract]</Label></Row><Row FlagID="0"><Id>2</Id><IsAbstractGroupTitle>false</IsAbstractGroupTitle><LabelSeparator>

</LabelSeparator><Level>2</Level><ElementName>us-gaap_LegalMattersAndContingenciesTextBlock</ElementName><ElementPrefix>us-gaap_</ElementPrefix><IsBaseElement>true</IsBaseElement><BalanceType>na</BalanceType><PeriodType>duration</PeriodType><IsReportTitle>false</IsReportTitle><IsSegmentTitle>false</IsSegmentTitle><IsCalendarTitle>false</IsCalendarTitle><IsEquityPrevioslyReportedAsRow>false</IsEquityPrevioslyReportedAsRow><IsEquityAdjustmentRow>false</IsEquityAdjustmentRow><IsBeginningBalance>false</IsBeginningBalance><IsEndingBalance>false</IsEndingBalance><IsReverseSign>false</IsReverseSign><PreferredLabelRole>terseLabel</PreferredLabelRole><FootnoteIndexer /><Cells><Cell FlagID="0" ContextID="eol_PE10781---1310-K0007_STD_364_20130628_0" UnitID=""><Id>1</Id><IsNumeric>false</IsNumeric><IsRatio>false</IsRatio><DisplayZeroAsNone>false</DisplayZeroAsNone><NumericAmount>0</NumericAmount><RoundedNumericAmount>0</RoundedNumericAmount><NonNumbericText>&lt;div&gt;
 &lt;p style="MARGIN-TOP: 0px; MARGIN-BOTTOM: 0px"&gt;&lt;font style="FONT-FAMILY: Times New Roman" size="2"&gt;&lt;b&gt;Note&amp;#xA0;5.&amp;#xA0;Legal
 Proceedings&lt;/b&gt;&lt;/font&gt;&lt;/p&gt;
 &lt;!-- xbrl,body --&gt;
 &lt;p style="MARGIN-TOP: 6px; TEXT-INDENT: 4%; MARGIN-BOTTOM: 0px"&gt;
 &lt;font style="FONT-FAMILY: Times New Roman" size="2"&gt;When the
 Company becomes aware of a claim or potential claim, the Company
 assesses the likelihood of any loss or exposure. The Company
 discloses information regarding each material claim where the
 likelihood of a loss contingency is probable or reasonably
 possible. If a loss contingency is probable and the amount of the
 loss can be reasonably estimated, the Company records an accrual
 for the loss. In such cases, there may be an exposure to potential
 loss in excess of the amount accrued. Where a loss is not probable
 but is reasonably possible or where a loss in excess of the amount
 accrued is reasonably possible, the Company discloses an estimate
 of the amount of the loss or range of possible losses for the claim
 if a reasonable estimate can be made, unless the amount of such
 reasonably possible losses is not material to the Company&amp;#x2019;s
 financial position, results of operations or cash flows. Unless
 otherwise stated below, for each of the matters described below,
 the Company has either recorded an accrual for losses that are
 probable and reasonably estimable or has determined that, while a
 loss is reasonably possible (including potential losses in excess
 of the amounts accrued by the Company), a reasonable estimate of
 the amount of loss or range of possible losses with respect to the
 claim or in excess of amounts already accrued by the Company cannot
 be made. The ability to predict the ultimate outcome of such
 matters involves judgments, estimates and inherent uncertainties.
 The actual outcome of such matters could differ materially from
 management&amp;#x2019;s estimates.&lt;/font&gt;&lt;/p&gt;
 &lt;p style="MARGIN-TOP: 12px; TEXT-INDENT: 4%; MARGIN-BOTTOM: 0px"&gt;
 &lt;font style="FONT-FAMILY: Times New Roman" size="2"&gt;Solely for
 purposes of this footnote, &amp;#x201C;WD&amp;#x201D; refers to Western
 Digital Corporation or one or more of its subsidiaries prior to the
 acquisition of HGST, &amp;#x201C;HGST&amp;#x201D; refers to HGST or one or
 more of its subsidiaries as of the Closing Date, and &amp;#x201C;the
 Company&amp;#x201D; refers to Western Digital Corporation and all of its
 subsidiaries on a consolidated basis including HGST.&lt;/font&gt;&lt;/p&gt;
 &lt;p style="MARGIN-TOP: 18px; MARGIN-BOTTOM: 0px; MARGIN-LEFT: 2%"&gt;
 &lt;font style="FONT-FAMILY: Times New Roman" size="2"&gt;&lt;i&gt;Intellectual
 Property Litigation&lt;/i&gt;&lt;/font&gt;&lt;/p&gt;
 &lt;p style="MARGIN-TOP: 6px; TEXT-INDENT: 4%; MARGIN-BOTTOM: 0px"&gt;
 &lt;font style="FONT-FAMILY: Times New Roman" size="2"&gt;On
 June&amp;#xA0;20, 2008, plaintiff Convolve, Inc.
 (&amp;#x201C;Convolve&amp;#x201D;) filed a complaint in the Eastern District
 of Texas against WD, HGST, and one other company alleging
 infringement of U.S. Patent Nos. 6,314,473 and 4,916,635. The
 complaint sought unspecified monetary damages and injunctive
 relief. On October&amp;#xA0;10, 2008, Convolve amended its complaint to
 allege infringement of only the &amp;#x2018;473 patent. The &amp;#x2018;473
 patent allegedly relates to interface technology to select between
 certain modes of a disk drive&amp;#x2019;s operations relating to speed
 and noise. A trial in the matter began on July&amp;#xA0;18, 2011 and
 concluded on July&amp;#xA0;26, 2011 with a verdict against WD and HGST
 in an amount that is not material to the Company&amp;#x2019;s financial
 position, results of operations or cash flows. WD and HGST have
 filed post-trial motions challenging the verdict and will evaluate
 their options for appeal after the Court rules on the post-trial
 motions.&lt;/font&gt;&lt;/p&gt;
 &lt;p style="MARGIN-TOP: 12px; TEXT-INDENT: 4%; MARGIN-BOTTOM: 0px"&gt;
 &lt;font style="FONT-FAMILY: Times New Roman" size="2"&gt;On
 December&amp;#xA0;7, 2009, plaintiff Nazomi Communications
 (&amp;#x201C;Nazomi&amp;#x201D;) filed a complaint in the Eastern District of
 Texas against WD and seven other companies alleging infringement of
 U.S. Patent Nos. 7,080,362 and 7,225,436. Nazomi dismissed the
 Eastern District of Texas suit after filing a similar complaint in
 the Central District of California on February&amp;#xA0;8, 2010. The
 case was subsequently transferred to the Northern District of
 California on October&amp;#xA0;14, 2010. The complaint seeks injunctive
 relief and unspecified monetary damages, fees and costs. The
 asserted patents allegedly relate to processor cores capable of
 Java hardware acceleration. In August 2012, the Court dismissed WD
 on summary judgment for non-infringement. Nazomi filed a notice of
 appeal on January&amp;#xA0;16, 2013. WD intends to continue to defend
 itself vigorously in this matter.&lt;/font&gt;&lt;/p&gt;
 &lt;p style="MARGIN-TOP: 12px; TEXT-INDENT: 4%; MARGIN-BOTTOM: 0px"&gt;
 &lt;font style="FONT-FAMILY: Times New Roman" size="2"&gt;On
 November&amp;#xA0;10, 2010, plaintiff Rembrandt Data Storage
 (&amp;#x201C;Rembrandt&amp;#x201D;) filed a complaint in the Western District
 of Wisconsin against WD alleging infringement of U.S. Patent Nos.
 5,995,342 and 6,195,232. The complaint seeks injunctive relief and
 unspecified monetary damages, fees and costs. The asserted patents
 allegedly relate to specific thin film heads having solenoid coils.
 After a favorable claim construction ruling by the court, WD
 secured a stipulation from Rembrandt to dismiss the case. Rembrandt
 appealed the Court&amp;#x2019;s claim construction ruling, and the
 Federal Circuit issued a summary affirmance in favor of WD in
 December 2012. Rembrandt did not appeal this ruling and the case
 has closed.&lt;/font&gt;&lt;/p&gt;
 &lt;p style="MARGIN-TOP: 12px; TEXT-INDENT: 4%; MARGIN-BOTTOM: 0px"&gt;
 &lt;font style="FONT-FAMILY: Times New Roman" size="2"&gt;On
 August&amp;#xA0;1, 2011, plaintiff Guzik Technical Enterprises
 (&amp;#x201C;Guzik&amp;#x201D;) filed a complaint in the Northern District of
 California against WD and various of its subsidiaries alleging
 infringement of U.S. Patent Nos. 6,023,145 and 6,785,085, breach of
 contract and misappropriation of trade secrets. The complaint seeks
 injunctive relief and unspecified monetary damages, fees and costs.
 The patents asserted by Guzik allegedly relate to devices used to
 test hard disk drive heads and media. WD has filed counterclaims
 against Guzik or patent infringement of U.S. Patent Nos. 5,844,420;
 5,640,089; 6,891,696; and 7,480,116. The patents asserted by WD
 relate to devices and methods used in the testing of hard disk
 drive heads and media. WD intends to defend itself vigorously in
 this matter.&lt;/font&gt;&lt;/p&gt;
 &lt;p style="MARGIN-TOP: 12px; MARGIN-BOTTOM: 0px; FONT-SIZE: 1px"&gt;
 &amp;#xA0;&lt;/p&gt;
 &lt;p style="MARGIN-TOP: 0px; TEXT-INDENT: 4%; MARGIN-BOTTOM: 0px"&gt;
 &lt;font style="FONT-FAMILY: Times New Roman" size="2"&gt;On
 April&amp;#xA0;29, 2013, plaintiffs Charles C. Freeney III et al. filed
 a complaint in the Eastern District of Texas against WD alleging
 infringement of U.S. Patent No.&amp;#xA0;7,110,744. The complaint seeks
 injunctive relief and unspecified monetary damages, fees and costs.
 The patent allegedly relates to WD&amp;#x2019;s AC router products. WD
 intends to defend itself vigorously in this matter.&lt;/font&gt;&lt;/p&gt;
 &lt;p style="MARGIN-TOP: 18px; MARGIN-BOTTOM: 0px; MARGIN-LEFT: 2%"&gt;
 &lt;font style="FONT-FAMILY: Times New Roman" size="2"&gt;&lt;i&gt;Seagate
 Matter&lt;/i&gt;&lt;/font&gt;&lt;/p&gt;
 &lt;p style="MARGIN-TOP: 6px; TEXT-INDENT: 4%; MARGIN-BOTTOM: 0px"&gt;
 &lt;font style="FONT-FAMILY: Times New Roman" size="2"&gt;On
 October&amp;#xA0;4, 2006, plaintiff Seagate Technology LLC
 (&amp;#x201C;Seagate&amp;#x201D;) filed an action in the District Court of
 Hennepin County, Minnesota, naming as defendants WD and one of its
 now former employees previously employed by Seagate. The complaint
 in the action alleged claims based on supposed misappropriation of
 trade secrets and sought injunctive relief and unspecified monetary
 damages, fees and costs. On June&amp;#xA0;19, 2007, WD&amp;#x2019;s former
 employee filed a demand for arbitration with the American
 Arbitration Association. A motion to stay the litigation as against
 all defendants and to compel arbitration of all Seagate&amp;#x2019;s
 claims was granted on September&amp;#xA0;19, 2007. On
 September&amp;#xA0;23, 2010, Seagate filed a motion to amend its claims
 and add allegations based on the supposed misappropriation of
 additional confidential information, and the arbitrator granted
 Seagate&amp;#x2019;s motion. The arbitration hearing commenced on
 May&amp;#xA0;23, 2011 and concluded on July&amp;#xA0;11, 2011.&lt;/font&gt;&lt;/p&gt;
 &lt;p style="MARGIN-TOP: 12px; TEXT-INDENT: 4%; MARGIN-BOTTOM: 0px"&gt;
 &lt;font style="FONT-FAMILY: Times New Roman" size="2"&gt;On
 November&amp;#xA0;18, 2011, the sole arbitrator ruled in favor of WD in
 connection with five of the eight alleged trade secrets at issue,
 based on evidence that such trade secrets were known publicly at
 the time the former employee joined WD. Based on a determination
 that the employee had fabricated evidence, the arbitrator then
 concluded that WD had to know of the fabrications. As a sanction,
 the arbitrator precluded any evidence or defense by WD disputing
 the validity, misappropriation, or use of the three remaining
 alleged trade secrets by WD, and entered judgment in favor of
 Seagate with respect to such trade secrets. Using an unjust
 enrichment theory of damages, the arbitrator issued an interim
 award against WD in the amount of $525 million plus pre-award
 interest at the Minnesota statutory rate of 10%&amp;#xA0;per year. In
 his decision with respect to these three trade secrets, the
 arbitrator did not question the relevance, veracity or credibility
 of any of WD&amp;#x2019;s ten expert and fact witnesses (other than
 WD&amp;#x2019;s former employee), nor the authenticity of any other
 evidence WD presented. On January&amp;#xA0;23, 2012, the arbitrator
 issued a final award adding pre-award interest in the amount of
 $105.4 million for a total final award of $630.4 million. On
 January&amp;#xA0;23, 2012, WD filed a petition in the District Court of
 Hennepin County, Minnesota to have the final arbitration award
 vacated. A hearing on the petition to vacate was held on
 March&amp;#xA0;1, 2012.&lt;/font&gt;&lt;/p&gt;
 &lt;p style="MARGIN-TOP: 12px; TEXT-INDENT: 4%; MARGIN-BOTTOM: 0px"&gt;
 &lt;font style="FONT-FAMILY: Times New Roman" size="2"&gt;On
 October&amp;#xA0;12, 2012, the District Court of Hennepin County,
 Minnesota vacated, in full, the $630.4 million final arbitration
 award. Specifically, the Court confirmed the arbitration award with
 respect to each of the five trade secret claims that WD and the
 former employee had won at the arbitration and vacated the
 arbitration award with respect to the three trade secret claims
 that WD and the former employee had lost at the arbitration. The
 Court ordered that a rehearing be held concerning those three
 alleged trade secret claims before a new arbitrator.&lt;/font&gt;&lt;/p&gt;
 &lt;p style="MARGIN-TOP: 12px; TEXT-INDENT: 4%; MARGIN-BOTTOM: 0px"&gt;
 &lt;font style="FONT-FAMILY: Times New Roman" size="2"&gt;On
 October&amp;#xA0;30, 2012, Seagate initiated an appeal of the District
 Court&amp;#x2019;s decision with the Minnesota Court of Appeals. Oral
 arguments in the appeal were held on April&amp;#xA0;24, 2013. On
 July&amp;#xA0;22, 2013, the Minnesota Court of Appeals reversed the
 District Court&amp;#x2019;s decision and remanded for entry of an order
 and judgment confirming the arbitration award. The Company strongly
 disagrees with the decision of the Court of Appeals, believes that
 the District Court&amp;#x2019;s decision was correct, and will file a
 petition for review with the Minnesota Supreme Court. If the
 Minnesota Supreme Court elects not to hear the Company&amp;#x2019;s
 petition for review or affirms the Court of Appeals decision, the
 District Court is expected to enter an order and judgment
 confirming the $630.4 million final arbitration award, plus
 post-award interest on the $525 million initial award at the
 statutory rate of 10% from January&amp;#xA0;24, 2012. No judgment will
 be entered while the Company is petitioning the Minnesota Supreme
 Court. The Company will continue to vigorously defend this matter.
 Nevertheless, in light of uncertainties, including the fact that
 the review of the Court of Appeals decision by the Minnesota
 Supreme Court is discretionary, the Company recorded an accrual of
 $681 million for this matter in its financial statements for the
 three months ended June 28, 2013. This amount is in addition to the
 $25 million previously accrued in the fourth quarter of fiscal
 2011. The total amount accrued of $706&amp;#xA0;million represents the
 amount of the final arbitration award, plus interest accrued on the
 initial arbitration award at the statutory rate of 10% from January
 24, 2012 through June 28, 2013.&lt;/font&gt;&lt;/p&gt;
 &lt;p style="MARGIN-TOP: 18px; MARGIN-BOTTOM: 0px; MARGIN-LEFT: 2%"&gt;
 &lt;font style="FONT-FAMILY: Times New Roman" size="2"&gt;&lt;i&gt;Other
 Matters&lt;/i&gt;&lt;/font&gt;&lt;/p&gt;
 &lt;p style="MARGIN-TOP: 6px; TEXT-INDENT: 4%; MARGIN-BOTTOM: 0px"&gt;
 &lt;font style="FONT-FAMILY: Times New Roman" size="2"&gt;In the normal
 course of business, the Company is subject to other legal
 proceedings, lawsuits and other claims. Although the ultimate
 aggregate amount of probable monetary liability or financial impact
 with respect to these other matters is subject to many
 uncertainties and is therefore not predictable with assurance,
 management believes that any monetary liability or financial impact
 to the Company from these other matters, individually and in the
 aggregate, would not be material to the Company&amp;#x2019;s financial
 condition, results of operations or cash flows. However, there can
 be no assurance with respect to such result, and monetary liability
 or financial impact to the Company from these other matters could
 differ materially from those projected.&lt;/font&gt;&lt;/p&gt;


 &lt;/div&gt;</NonNumbericText><FootnoteIndexer /><CurrencyCode /><CurrencySymbol /><IsIndependantCurrency>false</IsIndependantCurrency><ShowCurrencySymbol>false</ShowCurrencySymbol><DisplayDateInUSFormat>false</DisplayDateInUSFormat></Cell></Cells><ElementDataType>nonnum:textBlockItemType</ElementDataType><SimpleDataType>na</SimpleDataType><ElementDefenition>The entire disclosure for legal proceedings, legal contingencies, litigation, regulatory and environmental matters and other contingencies.</ElementDefenition><ElementReferences>No definition available.</ElementReferences><IsTotalLabel>false</IsTotalLabel><UnitID>0</UnitID><Label>Legal Proceedings</Label></Row></Rows><Footnotes /><IsEquityReport>false</IsEquityReport><ReportName>Legal Proceedings</ReportName><MonetaryRoundingLevel>UnKnown</MonetaryRoundingLevel><SharesRoundingLevel>UnKnown</SharesRoundingLevel><PerShareRoundingLevel>UnKnown</PerShareRoundingLevel><ExchangeRateRoundingLevel>UnKnown</ExchangeRateRoundingLevel><HasCustomUnits>true</HasCustomUnits><IsEmbedReport>false</IsEmbedReport><IsMultiCurrency>false</IsMultiCurrency><ReportType>Sheet</ReportType><RoleURI>http://www.westerndigital.com/taxonomy/role/NotesToFinancialStatementsLegalMattersAndContingenciesTextBlock</RoleURI><NumberOfCols>1</NumberOfCols><NumberOfRows>2</NumberOfRows></InstanceReport>
