<?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>1070 - Disclosure - FAIR VALUES OF ASSETS AND LIABILITIES</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>D2013Q2YTD</ContextID><EntitySchema>http://www.sec.gov/CIK</EntitySchema><EntityValue>0001027207</EntityValue><PeriodDisplayName /><PeriodType>duration</PeriodType><PeriodStartDate>2013-01-01T00:00:00</PeriodStartDate><PeriodEndDate>2013-06-30T00:00:00</PeriodEndDate><Segments /><Scenarios /></contextRef><UPS /><CurrencyCode /><OriginalCurrencyCode /></MCU><CurrencySymbol /><Labels><Label Key="CalendarSupplement" Id="0" Label="6 Months Ended" /><Label Key="Calendar" Id="1" Label="Jun. 30, 2013" /></Labels></Column></Columns><Rows><Row FlagID="0"><Id>1</Id><IsAbstractGroupTitle>true</IsAbstractGroupTitle><LabelSeparator>

</LabelSeparator><Level>1</Level><ElementName>us-gaap_FairValueDisclosuresAbstract</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>FAIR VALUES OF ASSETS AND LIABILITIES</Label></Row><Row FlagID="0"><Id>2</Id><IsAbstractGroupTitle>false</IsAbstractGroupTitle><LabelSeparator>

</LabelSeparator><Level>2</Level><ElementName>us-gaap_FairValueDisclosuresTextBlock</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="D2013Q2YTD" UnitID=""><Id>1</Id><IsNumeric>false</IsNumeric><IsRatio>false</IsRatio><DisplayZeroAsNone>false</DisplayZeroAsNone><NumericAmount>0</NumericAmount><RoundedNumericAmount>0</RoundedNumericAmount><NonNumbericText>&lt;div style="font-size:10.0pt;font-family:Times New Roman;"&gt;
&lt;p style="MARGIN: 0in 0in 0pt;"&gt;&lt;b&gt;&lt;font style="FONT-FAMILY: Times New Roman; FONT-SIZE: 10pt; FONT-WEIGHT: bold;" size="2"&gt;NOTE 7:&amp;#160;&amp;#160;&amp;#160; FAIR VALUES OF ASSETS AND LIABILITIES&lt;/font&gt;&lt;/b&gt;&lt;/p&gt;
&lt;p style="MARGIN: 0in 0in 0pt;"&gt;&amp;#160;&lt;/p&gt;
&lt;p style="MARGIN: 0in 0in 0pt;"&gt;&lt;font style="FONT-FAMILY: Times New Roman; FONT-SIZE: 10pt;" size="2"&gt;Accounting standards establish a three-level fair value hierarchy based upon the assumptions (inputs) used to price assets or liabilities. The hierarchy requires us to maximize the use of observable inputs and minimize the use of unobservable inputs. The three levels of inputs used to measure fair value are listed below.&lt;/font&gt;&lt;/p&gt;
&lt;p style="MARGIN: 0in 0in 0pt;"&gt;&amp;#160;&lt;/p&gt;
&lt;p style="TEXT-INDENT: 0.5in; MARGIN: 0in 0in 0pt;"&gt;&lt;font style="FONT-FAMILY: Times New Roman; FONT-SIZE: 10pt;" size="2"&gt;Level 1 &amp;#8212; Unadjusted quoted prices in active markets for identical assets or liabilities.&lt;/font&gt;&lt;/p&gt;
&lt;p style="MARGIN: 0in 0in 0pt;"&gt;&amp;#160;&lt;/p&gt;
&lt;p style="MARGIN: 0in 0in 0pt 0.5in;"&gt;&lt;font style="FONT-FAMILY: Times New Roman; FONT-SIZE: 10pt;" size="2"&gt;Level 2 &amp;#8212; Observable inputs other than those included in Level 1, such as quoted market prices for similar assets and liabilities in active markets or quoted prices for identical assets in inactive markets.&lt;/font&gt;&lt;/p&gt;
&lt;p style="MARGIN: 0in 0in 0pt 0.5in;"&gt;&amp;#160;&lt;/p&gt;
&lt;p style="MARGIN: 0in 0in 0pt 0.5in;"&gt;&lt;font style="FONT-FAMILY: Times New Roman; FONT-SIZE: 10pt;" size="2"&gt;Level 3 &amp;#8212; Unobservable inputs reflecting our own assumptions and best estimate of what inputs market participants would use in pricing an asset or liability.&lt;/font&gt;&lt;/p&gt;
&lt;p style="MARGIN: 0in 0in 0pt 0.5in;"&gt;&amp;#160;&lt;/p&gt;
&lt;p style="MARGIN: 0in 0in 0pt;"&gt;&lt;b&gt;&lt;i&gt;&lt;font style="FONT-STYLE: italic; FONT-FAMILY: Times New Roman; FONT-SIZE: 10pt; FONT-WEIGHT: bold;" size="2"&gt;Assets and Liabilities Measured at Fair Value on a Recurring Basis&lt;/font&gt;&lt;/i&gt;&lt;/b&gt;&lt;/p&gt;
&lt;p style="MARGIN: 0in 0in 0pt;"&gt;&amp;#160;&lt;/p&gt;
&lt;p style="MARGIN: 0in 0in 0pt;"&gt;&lt;font style="FONT-FAMILY: Times New Roman; FONT-SIZE: 10pt;" size="2"&gt;The carrying value and estimated fair value of our cash equivalents, which consist of short-term money market funds, are classified as Level 1. There have been no transfers between Level 1 and Level 2 during the six months ended June&amp;#160;30, 2013.&amp;#160; In connection with the acquisitions of De Novo and Jupiter eSources we established liabilities related to potential contingent consideration that were considered to be Level 3 liabilities.&amp;#160; These liabilities were valued using unobservable inputs to the valuation methodology that are significant to the measurement of the fair value. As of June&amp;#160;30, 2013 and December&amp;#160;31, 2012, the value of these liabilities was $0.&lt;/font&gt;&lt;/p&gt;
&lt;p style="MARGIN: 0in 0in 0pt;"&gt;&amp;#160;&lt;/p&gt;
&lt;p style="MARGIN: 0in 0in 0pt;"&gt;&lt;font style="FONT-FAMILY: Times New Roman; FONT-SIZE: 10pt;" size="2"&gt;For fair value measurements categorized within Level 3 of the fair value hierarchy, our accounting and finance management, who report to the chief financial officer, determine our valuation policies and procedures.&amp;#160; The development and determination of the unobservable inputs for Level 3 fair value measurements and fair value calculations are the responsibility of our accounting and finance management and are approved by the chief financial officer.&amp;#160; Fair value calculations are generally prepared by third-party valuation experts who rely on discussions with management in addition to the use of management&amp;#8217;s assumptions and estimates as they relate to the assets or liabilities in Level 3.&amp;#160; Such assumptions and estimates include such inputs as estimates of future cash flows, projected profit and loss information, discount rates, and assumptions as they relate to future pertinent events.&amp;#160; Through regular interaction with the third-party valuation experts, finance and accounting management determine that the valuation techniques used and inputs and outputs of the models reflect the requirements of accounting standards as they relate to fair value measurements.&amp;#160; Changes in fair value measurements categorized within Level 3 of the fair value hierarchy are analyzed each period based on changes in estimates or assumptions and recorded as appropriate.&amp;#160; As of June&amp;#160;30, 2013 there were no assets or liabilities that are measured and recorded at fair value on a recurring basis.&lt;/font&gt;&lt;/p&gt;
&lt;p style="MARGIN: 0in 0in 0pt;"&gt;&amp;#160;&lt;/p&gt;
&lt;p style="MARGIN: 0in 0in 0pt;"&gt;&lt;b&gt;&lt;i&gt;&lt;font style="FONT-STYLE: italic; FONT-FAMILY: Times New Roman; FONT-SIZE: 10pt; FONT-WEIGHT: bold;" size="2"&gt;Fair Value of Financial Assets and Liabilities&lt;/font&gt;&lt;/i&gt;&lt;/b&gt;&lt;/p&gt;
&lt;p style="MARGIN: 0in 0in 0pt;"&gt;&amp;#160;&lt;/p&gt;
&lt;p style="MARGIN: 0in 0in 0pt;"&gt;&lt;font style="FONT-FAMILY: Times New Roman; FONT-SIZE: 10pt;" size="2"&gt;As of June&amp;#160;30, 2013 and December&amp;#160;31, 2012, the carrying value of our trade accounts receivable, accounts payable, certain other liabilities, deferred acquisition price payments and capital leases approximated fair value. The amount outstanding under our credit facility was $238.0 million and $199.0 million at June&amp;#160;30, 2013 and December&amp;#160;31, 2012, respectively, which approximated fair value due to the borrowing rates currently available to us for debt with similar terms and is classified as Level 2.&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 the fair value of financial instruments (as defined), including financial assets and financial liabilities (collectively, as defined), and the measurements of those instruments as well as disclosures related to the fair value of non-financial assets and liabilities. Such disclosures about the financial instruments, assets, and liabilities would include: (1) the fair value of the required items together with their carrying amounts (as appropriate); (2) for items for which it is not practicable to estimate fair value, disclosure would include: (a) information pertinent to estimating fair value (including, carrying amount, effective interest rate, and maturity, and (b) the reasons why it is not practicable to estimate fair value; (3) significant concentrations of credit risk including: (a) information about the activity, region, or economic characteristics identifying a concentration, (b) the maximum amount of loss the entity is exposed to based on the gross fair value of the related item, (c) policy for requiring collateral or other security and information as to accessing such collateral or security, and (d) the nature and brief description of such collateral or security; (4) quantitative information about market risks and how such risks are managed; (5) for items measured on both a recurring and nonrecurring basis information regarding the inputs used to develop the fair value measurement; and (6) for items presented in the financial statement for which fair value measurement is elected: (a) information necessary to understand the reasons for the election, (b) discussion of the effect of fair value changes on earnings, (c) a description of [similar groups] items for which the election is made and the relation thereof to the balance sheet, the aggregate carrying value of items included in the balance sheet that are not eligible for the election; (7) all other required (as defined) and desired information.</ElementDefenition><ElementReferences>Reference 1: http://www.xbrl.org/2003/role/presentationRef

 -Publisher FASB

 -Name Statement of Financial Accounting Standard (FAS)

 -Number 107

 -Paragraph 15A

 -Subparagraph a-d

 -LegacyDoc This reference is SUPERSEDED by the Accounting Standards Codification effective for interim and annual periods ending after September 15, 2009.  This reference is included to help users transition from the previous accounting hierarchy and will be removed from future versions of this taxonomy.



Reference 2: http://www.xbrl.org/2003/role/presentationRef

 -Publisher FASB

 -Name Statement of Financial Accounting Standard (FAS)

 -Number 107

 -Paragraph 15C, 15D

 -LegacyDoc This reference is SUPERSEDED by the Accounting Standards Codification effective for interim and annual periods ending after September 15, 2009.  This reference is included to help users transition from the previous accounting hierarchy and will be removed from future versions of this taxonomy.



Reference 3: http://www.xbrl.org/2003/role/presentationRef

 -Publisher FASB

 -Name Statement of Financial Accounting Standard (FAS)

 -Number 107

 -Paragraph 3, 10, 14, 15

 -LegacyDoc This reference is SUPERSEDED by the Accounting Standards Codification effective for interim and annual periods ending after September 15, 2009.  This reference is included to help users transition from the previous accounting hierarchy and will be removed from future versions of this taxonomy.



Reference 4: http://www.xbrl.org/2003/role/presentationRef

 -Publisher FASB

 -Name Statement of Financial Accounting Standard (FAS)

 -Number 157

 -Paragraph 32, 33, 34

 -LegacyDoc This reference is SUPERSEDED by the Accounting Standards Codification effective for interim and annual periods ending after September 15, 2009.  This reference is included to help users transition from the previous accounting hierarchy and will be removed from future versions of this taxonomy.



Reference 5: http://www.xbrl.org/2003/role/presentationRef

 -Publisher FASB

 -Name Accounting Standards Codification

 -Topic 825

 -SubTopic 10

 -Section 50

 -Paragraph 21

 -URI http://asc.fasb.org/extlink&amp;oid=7491637&amp;loc=d3e13537-108611



Reference 6: http://www.xbrl.org/2003/role/presentationRef

 -Publisher FASB

 -Name Statement of Financial Accounting Standard (FAS)

 -Number 133

 -Paragraph 44A, 44B

 -LegacyDoc This reference is SUPERSEDED by the Accounting Standards Codification effective for interim and annual periods ending after September 15, 2009.  This reference is included to help users transition from the previous accounting hierarchy and will be removed from future versions of this taxonomy.



Reference 7: http://www.xbrl.org/2003/role/presentationRef

 -Publisher FASB

 -Name Accounting Standards Codification

 -Topic 825

 -SubTopic 10

 -Section 50

 -Paragraph 10

 -URI http://asc.fasb.org/extlink&amp;oid=7491637&amp;loc=d3e13433-108611



Reference 8: http://www.xbrl.org/2003/role/presentationRef

 -Publisher FASB

 -Name Accounting Standards Codification

 -Topic 825

 -SubTopic 10

 -Section 50

 -Paragraph 28

 -URI http://asc.fasb.org/extlink&amp;oid=6957238&amp;loc=d3e14064-108612



Reference 9: http://www.xbrl.org/2003/role/presentationRef

 -Publisher FASB

 -Name Accounting Standards Codification

 -Topic 820

 -SubTopic 10

 -Section 50

 -Paragraph 2

 -URI http://asc.fasb.org/extlink&amp;oid=7578670&amp;loc=d3e19207-110258



Reference 10: http://www.xbrl.org/2003/role/presentationRef

 -Publisher FASB

 -Name Accounting Standards Codification

 -Topic 825

 -SubTopic 10

 -Section 50

 -Paragraph 30

 -URI http://asc.fasb.org/extlink&amp;oid=6957238&amp;loc=d3e14172-108612



Reference 11: http://www.xbrl.org/2003/role/presentationRef

 -Publisher FASB

 -Name Accounting Standards Codification

 -Topic 825

 -SubTopic 10

 -Section 50

 -Paragraph 16

 -URI http://asc.fasb.org/extlink&amp;oid=7491637&amp;loc=d3e13504-108611



Reference 12: http://www.xbrl.org/2003/role/presentationRef

 -Publisher FASB

 -Name Statement of Financial Accounting Standard (FAS)

 -Number 107

 -Paragraph 15B

 -Subparagraph a, b

 -LegacyDoc This reference is SUPERSEDED by the Accounting Standards Codification effective for interim and annual periods ending after September 15, 2009.  This reference is included to help users transition from the previous accounting hierarchy and will be removed from future versions of this taxonomy.



Reference 13: http://www.xbrl.org/2003/role/presentationRef

 -Publisher FASB

 -Name Statement of Financial Accounting Standard (FAS)

 -Number 159

 -Paragraph 17-22, 27, 28

 -LegacyDoc This reference is SUPERSEDED by the Accounting Standards Codification effective for interim and annual periods ending after September 15, 2009.  This reference is included to help users transition from the previous accounting hierarchy and will be removed from future versions of this taxonomy.



</ElementReferences><IsTotalLabel>false</IsTotalLabel><UnitID>0</UnitID><Label>FAIR VALUES OF ASSETS AND LIABILITIES</Label></Row></Rows><Footnotes /><IsEquityReport>false</IsEquityReport><ReportName>FAIR VALUES OF ASSETS AND LIABILITIES</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.epiqsystems.com/role/DisclosureFairValuesOfAssetsAndLiabilities</RoleURI><NumberOfCols>1</NumberOfCols><NumberOfRows>2</NumberOfRows></InstanceReport>
