<?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>111 - Disclosure - Impairment of Goodwill and Long-Lived Assets</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>P10_01_2012To06_30_2013</ContextID><EntitySchema>http://www.sec.gov/CIK</EntitySchema><EntityValue>0000030831</EntityValue><PeriodDisplayName /><PeriodType>duration</PeriodType><PeriodStartDate>2012-10-01T00:00:00</PeriodStartDate><PeriodEndDate>2013-06-30T00:00:00</PeriodEndDate><Segments /><Scenarios /></contextRef><UPS /><CurrencyCode /><OriginalCurrencyCode /></MCU><CurrencySymbol /><Labels><Label Key="CalendarSupplement" Id="0" Label="9 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_AssetImpairmentChargesAbstract</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>Asset Impairment Charges [Abstract]</Label></Row><Row FlagID="0"><Id>2</Id><IsAbstractGroupTitle>false</IsAbstractGroupTitle><LabelSeparator>

</LabelSeparator><Level>2</Level><ElementName>us-gaap_AssetImpairmentChargesTextBlock</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>verboseLabel</PreferredLabelRole><FootnoteIndexer /><Cells><Cell FlagID="0" ContextID="P10_01_2012To06_30_2013" UnitID=""><Id>1</Id><IsNumeric>false</IsNumeric><IsRatio>false</IsRatio><DisplayZeroAsNone>false</DisplayZeroAsNone><NumericAmount>0</NumericAmount><RoundedNumericAmount>0</RoundedNumericAmount><NonNumbericText>              &lt;table border="0" style="clear:both;width:100%; table-layout:fixed;"&gt;  &lt;tr&gt;  &lt;td&gt;&lt;/td&gt;  &lt;/tr&gt;  &lt;/table&gt;    &lt;div style="clear:both; FONT-FAMILY:Times New Roman;FONT-SIZE: 10pt;MARGIN: 0pt 0px; FONT: 10pt Times New Roman, Times, Serif "&gt;  &lt;div style="clear:both; FONT-FAMILY:Times New Roman;FONT-SIZE: 10pt;MARGIN: 0pt 0px; FONT: 10pt Times New Roman, Times, Serif"&gt;  &lt;b&gt;Note 5 &amp;#150; Impairment of Goodwill and Long-Lived  Assets&lt;/b&gt;&lt;/div&gt;    &lt;div style="clear:both; FONT-FAMILY:Times New Roman;FONT-SIZE: 10pt"&gt;  &amp;#160;&lt;/div&gt;    &lt;div style="clear:both; FONT-FAMILY:Times New Roman;FONT-SIZE: 10pt;TEXT-ALIGN: justify; MARGIN: 0pt 0px; FONT: 10pt Times New Roman, Times, Serif"&gt;  Goodwill is subject to an annual impairment test. The Company  considers many factors which may indicate the requirement to  perform additional, interim impairment tests. These include:&lt;/div&gt;    &lt;div style="clear:both; FONT-FAMILY:Times New Roman;FONT-SIZE: 10pt"&gt;  &amp;#160;&lt;/div&gt;    &lt;table style="clear:both;MARGIN-TOP: 0pt; FONT: 10pt Times New Roman, Times, Serif; MARGIN-BOTTOM: 0pt"   cellspacing="0" cellpadding="0" width="100%"&gt;  &lt;tr style="VERTICAL-ALIGN: top"&gt;  &lt;td style="WIDTH: 24pt"&gt;&lt;/td&gt;  &lt;td style="WIDTH: 18pt"&gt;  &lt;div&gt;&lt;font style="FONT-FAMILY: Symbol"&gt;&amp;#8901;&lt;/font&gt;&lt;/div&gt;  &lt;/td&gt;  &lt;td style="TEXT-ALIGN: justify"&gt;  &lt;div&gt;&lt;font style="FONT-FAMILY: Times New Roman, Times, Serif"&gt;A  significant adverse long term outlook for any of our  industries;&lt;/font&gt;&lt;/div&gt;  &lt;/td&gt;  &lt;/tr&gt;  &lt;/table&gt;    &lt;table style="clear:both;MARGIN-TOP: 0pt; FONT: 10pt Times New Roman, Times, Serif; MARGIN-BOTTOM: 0pt"   cellspacing="0" cellpadding="0" width="100%"&gt;  &lt;tr style="VERTICAL-ALIGN: top"&gt;  &lt;td style="WIDTH: 24pt"&gt;&lt;/td&gt;  &lt;td style="WIDTH: 18pt"&gt;  &lt;div&gt;&lt;font style="FONT-FAMILY: Symbol"&gt;&amp;#8901;&lt;/font&gt;&lt;/div&gt;  &lt;/td&gt;  &lt;td style="TEXT-ALIGN: justify"&gt;  &lt;div&gt;&lt;font style="FONT-FAMILY: Times New Roman, Times, Serif"&gt;An  adverse finding or rejection from a regulatory body involved in new  product regulatory approvals;&lt;/font&gt;&lt;/div&gt;  &lt;/td&gt;  &lt;/tr&gt;  &lt;/table&gt;    &lt;table style="clear:both;MARGIN-TOP: 0pt; FONT: 10pt Times New Roman, Times, Serif; MARGIN-BOTTOM: 0pt"   cellspacing="0" cellpadding="0" width="100%"&gt;  &lt;tr style="VERTICAL-ALIGN: top"&gt;  &lt;td style="WIDTH: 24pt"&gt;&lt;/td&gt;  &lt;td style="WIDTH: 18pt"&gt;  &lt;div&gt;&lt;font style="FONT-FAMILY: Symbol"&gt;&amp;#8901;&lt;/font&gt;&lt;/div&gt;  &lt;/td&gt;  &lt;td style="TEXT-ALIGN: justify"&gt;  &lt;div&gt;&lt;font style="FONT-FAMILY: Times New Roman, Times, Serif"&gt;  Failure of an anticipated commercialization of a product or product  line;&lt;/font&gt;&lt;/div&gt;  &lt;/td&gt;  &lt;/tr&gt;  &lt;/table&gt;    &lt;table style="clear:both;MARGIN-TOP: 0pt; FONT: 10pt Times New Roman, Times, Serif; MARGIN-BOTTOM: 0pt"   cellspacing="0" cellpadding="0" width="100%"&gt;  &lt;tr style="VERTICAL-ALIGN: top"&gt;  &lt;td style="WIDTH: 24pt"&gt;&lt;/td&gt;  &lt;td style="WIDTH: 18pt"&gt;  &lt;div&gt;&lt;font style="FONT-FAMILY: Symbol"&gt;&amp;#8901;&lt;/font&gt;&lt;/div&gt;  &lt;/td&gt;  &lt;td style="TEXT-ALIGN: justify"&gt;  &lt;div&gt;&lt;font style="FONT-FAMILY: Times New Roman, Times, Serif"&gt;  Unanticipated competition or the introduction of a disruptive  technology;&lt;/font&gt;&lt;/div&gt;  &lt;/td&gt;  &lt;/tr&gt;  &lt;/table&gt;    &lt;table style="clear:both;MARGIN-TOP: 0pt; FONT: 10pt Times New Roman, Times, Serif; MARGIN-BOTTOM: 0pt"   cellspacing="0" cellpadding="0" width="100%"&gt;  &lt;tr style="VERTICAL-ALIGN: top"&gt;  &lt;td style="WIDTH: 24pt"&gt;&lt;/td&gt;  &lt;td style="WIDTH: 18pt"&gt;  &lt;div&gt;&lt;font style="FONT-FAMILY: Symbol"&gt;&amp;#8901;&lt;/font&gt;&lt;/div&gt;  &lt;/td&gt;  &lt;td&gt;  &lt;div&gt;&lt;font style="FONT-FAMILY: Times New Roman, Times, Serif"&gt;The  testing for recoverability under the Impairment or Disposal of  Long-Lived Assets Subsections of Subtopic 360-10 of a significant  asset group within a reporting unit;&lt;/font&gt;&lt;/div&gt;  &lt;/td&gt;  &lt;/tr&gt;  &lt;/table&gt;    &lt;table style="clear:both;MARGIN-TOP: 0pt; FONT: 10pt Times New Roman, Times, Serif; MARGIN-BOTTOM: 0pt"   cellspacing="0" cellpadding="0" width="100%"&gt;  &lt;tr style="VERTICAL-ALIGN: top"&gt;  &lt;td style="WIDTH: 24pt"&gt;&lt;/td&gt;  &lt;td style="WIDTH: 18pt"&gt;  &lt;div&gt;&lt;font style="FONT-FAMILY: Symbol"&gt;&amp;#8901;&lt;/font&gt;&lt;/div&gt;  &lt;/td&gt;  &lt;td&gt;  &lt;div&gt;&lt;font style="FONT-FAMILY: Times New Roman, Times, Serif"&gt;A  loss of key personnel; and&lt;/font&gt;&lt;/div&gt;  &lt;/td&gt;  &lt;/tr&gt;  &lt;/table&gt;    &lt;table style="clear:both;MARGIN-TOP: 0pt; FONT: 10pt Times New Roman, Times, Serif; MARGIN-BOTTOM: 0pt"   cellspacing="0" cellpadding="0" width="100%"&gt;  &lt;tr style="VERTICAL-ALIGN: top"&gt;  &lt;td style="WIDTH: 24pt"&gt;&lt;/td&gt;  &lt;td style="WIDTH: 18pt"&gt;  &lt;div&gt;&lt;font style="FONT-FAMILY: Symbol"&gt;&amp;#8901;&lt;/font&gt;&lt;/div&gt;  &lt;/td&gt;  &lt;td&gt;  &lt;div&gt;&lt;font style="FONT-FAMILY: Times New Roman, Times, Serif"&gt;An  expectation that a reporting unit carrying goodwill, or a  significant portion of a reporting unit, will be sold or otherwise  disposed of.&lt;/font&gt;&lt;/div&gt;  &lt;/td&gt;  &lt;/tr&gt;  &lt;/table&gt;    &lt;div style="clear:both; FONT-FAMILY:Times New Roman;FONT-SIZE: 10pt"&gt;  &amp;#160;&lt;/div&gt;    &lt;div style="clear:both; FONT-FAMILY:Times New Roman;FONT-SIZE: 10pt;TEXT-ALIGN: justify; MARGIN: 0pt 0px; FONT: 10pt Times New Roman, Times, Serif"&gt;  The Company&amp;#8217;s Dynasil Products subsidiary  (&amp;#8220;Products&amp;#8221;) has experienced delays in obtaining  regulatory approval of its next generation lead paint analyzer  product. In the second quarter of fiscal 2013, due to the  uncertainties regarding whether approval of this updated product  could be obtained in a reasonable timeframe, Products determined it  would limit its activities to the sale and service of its current  lead paint analyzer. The Company is currently evaluating various  strategic alternatives associated with the lead paint analyzer  product.&lt;/div&gt;    &lt;div style="clear:both; FONT-FAMILY:Times New Roman;FONT-SIZE: 10pt"&gt;  &amp;#160;&lt;/div&gt;    &lt;div style="clear:both; FONT-FAMILY:Times New Roman;FONT-SIZE: 10pt;TEXT-ALIGN: justify; MARGIN: 0pt 0px; FONT: 10pt Times New Roman, Times, Serif"&gt;  Products also experienced delays in obtaining regulatory approvals  of its updated medical probe for cancer surgery. In May 2013,  Products received approval for the sale of its updated medical  probe for cancer surgery in the United States and began shipping  its new &lt;i&gt;Navigator 2.0&amp;#153;&lt;/i&gt; units. Products has begun  testing the probe for sterility after which it expects to obtain  the CE mark and other certifications necessary for sales to various  countries outside the United States.&lt;/div&gt;    &lt;div style="clear:both; FONT-FAMILY:Times New Roman;FONT-SIZE: 10pt"&gt;  &amp;#160;&lt;/div&gt;    &lt;div style="clear:both; FONT-FAMILY:Times New Roman;FONT-SIZE: 10pt;TEXT-ALIGN: justify; MARGIN: 0pt 0px; FONT: 10pt Times New Roman, Times, Serif"&gt;  As a result of the downsizing of expenditures associated with both  product lines and the process of evaluating strategic alternatives  associated with the lead paint analyzer product during the second  quarter of fiscal 2013, the Company determined that an interim  impairment test of long lived assets and goodwill associated with  this business unit was required in the second quarter of 2013.  Based on its analyses, the Company determined that the fair value  of the long-lived assets (other than goodwill) of Products was less  than the carrying amount of those assets and, as a result, recorded  a non-cash, pre-tax impairment charge of approximately $2,748,000  during the quarter ended March 31, 2013. The Company then performed  a goodwill impairment test as prescribed in ASC 350 and recognized  a non-cash, pre-tax charge of approximately $4,015,000 during the  quarter ended March 31, 2013. The resulting write-offs, which total  approximately $6,763,000 are included in the Company&amp;#8217;s  results of operations for the nine months ended June 30,  2013.&lt;/div&gt;    &lt;div style="clear:both; FONT-FAMILY:Times New Roman;FONT-SIZE: 10pt"&gt;  &amp;#160;&lt;/div&gt;    &lt;div style="clear:both; FONT-FAMILY:Times New Roman;FONT-SIZE: 10pt;TEXT-ALIGN: justify; MARGIN: 0pt 0px; FONT: 10pt Times New Roman, Times, Serif"&gt;  The valuation methods utilized to value the long lived assets and  the goodwill discussed above are based on the amount and timing of  expected future cash flows and growth rates and include a  determination of an appropriate discount rate. The cash flows  utilized in the discounted cash flow analyses were based on  financial forecasts developed internally by management. Estimating  future cash flows requires significant judgment and projections may  vary from the cash flows eventually realized. The discount rate  used also requires significant judgment and is intended to reflect  the risks involved in the business.&lt;/div&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 details of the charge against earnings resulting from the aggregate write down of all assets from their carrying value to their fair value. Disclosure may also include a description of the impaired asset and facts and circumstances leading to the impairment, amount of the impairment loss and where the loss is located in the income statement, method(s) for determining fair value, and the segment in which the impaired asset is reported.</ElementDefenition><ElementReferences>No definition available.</ElementReferences><IsTotalLabel>false</IsTotalLabel><UnitID>0</UnitID><Label>Asset Impairment Charges [Text Block]</Label></Row></Rows><Footnotes /><IsEquityReport>false</IsEquityReport><ReportName>Impairment of Goodwill and Long-Lived Assets</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.dynasilcorp.com/role/ImpairmentOfGoodwillAndLonglivedAssets</RoleURI><NumberOfCols>1</NumberOfCols><NumberOfRows>2</NumberOfRows></InstanceReport>
