<?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>2104100 - Disclosure - Liquidity and Continued Operations</ReportLongName><DisplayLabelColumn>true</DisplayLabelColumn><ShowElementNames>false</ShowElementNames><RoundingOption /><HasEmbeddedReports>false</HasEmbeddedReports><Columns><Column FlagID="0"><Id>1</Id><IsAbstractGroupTitle>false</IsAbstractGroupTitle><LabelSeparator>

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</LabelSeparator><Level>1</Level><ElementName>asti_LiquidityAndContinuedOperationsAbstract</ElementName><ElementPrefix>asti_</ElementPrefix><IsBaseElement>false</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>LIQUIDITY AND CONTINUED OPERATIONS [Abstract]</Label></Row><Row FlagID="0"><Id>2</Id><IsAbstractGroupTitle>false</IsAbstractGroupTitle><LabelSeparator>

</LabelSeparator><Level>2</Level><ElementName>asti_LiquidityAndContinuedOperationsTextBlock</ElementName><ElementPrefix>asti_</ElementPrefix><IsBaseElement>false</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-family:Times New Roman;font-size:10pt;"&gt;&lt;div style="line-height:120%;font-size:10pt;"&gt;&lt;font style="font-family:inherit;font-size:10pt;font-weight:bold;"&gt;LIQUIDITY AND CONTINUED OPERATIONS&lt;/font&gt;&lt;/div&gt;&lt;div style="line-height:120%;padding-top:12px;text-indent:32px;font-size:10pt;"&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt;As of &lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;color:#000000;text-decoration:none;"&gt;June&amp;#160;30, 2013&lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt;, the Company had &lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;color:#000000;text-decoration:none;"&gt;$4.0 million&lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt; in cash and working capital of &lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;color:#000000;text-decoration:none;"&gt;$4.1 million&lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt;. As discussed in Note 2, the Company is in the development stage and is currently incurring significant losses from operations as it works toward commercialization. The Company made cash payments of &lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;color:#000000;text-decoration:none;"&gt;$0.6 million&lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt; in the &lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;color:#000000;text-decoration:none;"&gt;six&lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt; months ended &lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;color:#000000;text-decoration:none;"&gt;June&amp;#160;30, 2013&lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt; for property, plant and equipment. In May 2013, the Company completed the sale of &lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;color:#000000;text-decoration:none;"&gt;2,500,000&lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt; shares of common stock in a private placement for proceeds of &lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;color:#000000;text-decoration:none;"&gt;$1.4 million&lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt;. In June 2013, the Company contracted to sell &lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;color:#000000;text-decoration:none;"&gt;750,000&lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt; shares of Series A preferred stock and warrants to purchase up to &lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;color:#000000;text-decoration:none;"&gt;2,625,000&lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt; shares of common stock in a private placement for gross proceeds of &lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;color:#000000;text-decoration:none;"&gt;$6.0 million&lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt;. As of &lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;color:#000000;text-decoration:none;"&gt;June&amp;#160;30, 2013&lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt;, the Company had received &lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;color:#000000;text-decoration:none;"&gt;$1.0 million&lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt; from the sale of preferred stock and warrants. On August 7, 2013, the Company had a second closing and received &lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;color:#000000;text-decoration:none;"&gt;$3.0 million&lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt;. The third closing and the receipt of the final &lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;color:#000000;text-decoration:none;"&gt;$2.0 million&lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt; is expected to occur in August 2013.&lt;/font&gt;&lt;/div&gt;&lt;div style="line-height:120%;padding-top:12px;text-indent:32px;font-size:10pt;"&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt;The Company has commenced production at its manufacturing facility. The Company does not expect that sales revenue and cash flows will be sufficient to support operations and cash requirements until it has fully implemented its new consumer products strategy. Changes in the level of expected operating losses, the timing of planned capital expenditures or other factors may negatively impact cash flows and reduce current cash and investments faster than anticipated. During the second quarter of 2013, the Company used&amp;#160;&lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;color:#000000;text-decoration:none;"&gt;$4.9 million&lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt;&amp;#160;in cash for operations, as compared to the prior quarter when &lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;color:#000000;text-decoration:none;"&gt;$5.1 million&lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt; was used for operations. &lt;/font&gt;&lt;/div&gt;&lt;div style="line-height:120%;padding-top:12px;text-indent:32px;font-size:10pt;"&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt;In 2013, the Company expects to incur a base level of maintenance capital expenditures and relatively minor improvements to the existing asset base. The Company's primary significant long term obligation consists of a note payable of&amp;#160;&lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;color:#000000;text-decoration:none;"&gt;$6.5 million&lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt;&amp;#160;to a financial institution secured by a mortgage on its headquarters and manufacturing building in Thornton, Colorado. Total payments of&amp;#160;&lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;color:#000000;text-decoration:none;"&gt;$0.3 million&lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt;, including principal and interest, will come due during the remainder of 2013. Additional projected product revenues are not anticipated to result in a positive cash flow position for the year 2013 overall. Upon completion of the Series A and warrant financing, the Company will require minimal additional cash liquidity for the year ending December 31, 2013. The Company continues to accelerate sales and marketing efforts related to its consumer products strategy through increased hiring. The Company has begun activities related to securing additional financing through strategic or financial investors, but there is no assurance that the Company will be able to raise additional capital on acceptable terms or at all. If the Company's revenues do not increase rapidly, and/or additional financing is not obtained, the Company will be required to significantly curtail operations to reduce costs and/or sell assets. Such actions would likely have an adverse impact on the Company's future operations.&lt;/font&gt;&lt;/div&gt;&lt;div style="line-height:120%;padding-top:12px;text-indent:30px;font-size:10pt;"&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt;On April 11, 2012, the Company received notice from The NASDAQ Stock Market (&amp;#8220;Nasdaq&amp;#8221;) stating that because the Company had not regained compliance with the &lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;color:#000000;text-decoration:none;"&gt;$1.00&lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt; minimum bid price requirement for continued listing, the Company's common stock (listed on The Nasdaq Global Market) would be subject to delisting. On August 17, 2012, the Company received notification from The Nasdaq Listing Qualifications department that the Company had regained compliance with the minimum bid price requirement, and that the Company's noncompliance had been rectified. On December 5, 2012, the Company received notice from Nasdaq stating that it had again fallen out of compliance with the &lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;color:#000000;text-decoration:none;"&gt;$1.00&lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt;&amp;#160;minimum bid price requirement for continued listing. The Company was provided &lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;color:#000000;text-decoration:none;"&gt;180&lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt; days to regain compliance. On June 4, 2013, the Company received notice from Nasdaq stating that because the Company had not regained compliance with the minimum bid price requirement for continued listing, the Company's common stock would be subject to delisting. The Company presented its plan to regain compliance before a Nasdaq Hearings Panel on July 11, 2013. On August 5, 2013, the Company received notification from The Nasdaq Listing Qualifications department that the Company had regained compliance with the minimum bid price requirement, and that the Company's noncompliance had been rectified.&lt;/font&gt;&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>LIQUIDITY AND CONTINUED OPERATION</ElementDefenition><ElementReferences>No definition available.</ElementReferences><IsTotalLabel>false</IsTotalLabel><UnitID>0</UnitID><Label>Liquidity and Continued Operations</Label></Row></Rows><Footnotes /><IsEquityReport>false</IsEquityReport><ReportName>Liquidity and Continued Operations</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.ascentsolar.com/role/LiquidityAndContinuedOperations</RoleURI><NumberOfCols>1</NumberOfCols><NumberOfRows>2</NumberOfRows></InstanceReport>
