<?xml version="1.0" encoding="utf-8"?>
<InstanceReport xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:xsd="http://www.w3.org/2001/XMLSchema">
  <Version>2.2.0.7</Version>
  <hasSegments>false</hasSegments>
  <ReportName>Organization and Basis of Presentation</ReportName>
  <ReportLongName>0201 - Disclosure - Organization and Basis of Presentation</ReportLongName>
  <DisplayLabelColumn>true</DisplayLabelColumn>
  <ShowElementNames>false</ShowElementNames>
  <RoundingOption />
  <HasEmbeddedReports>false</HasEmbeddedReports>
  <Columns>
    <Column>
      <LabelColumn>false</LabelColumn>
      <Id>1</Id>
      <Labels>
        <Label Id="1" Label="6 Months Ended" />
        <Label Id="2" Label="Jun. 30, 2010" />
      </Labels>
      <CurrencyCode>USD</CurrencyCode>
      <FootnoteIndexer />
      <hasSegments>false</hasSegments>
      <hasScenarios>false</hasScenarios>
      <Segments />
      <Scenarios />
      <Units>
        <Unit>
          <UnitID>USD</UnitID>
          <UnitType>Standard</UnitType>
          <StandardMeasure>
            <MeasureSchema>http://www.xbrl.org/2003/iso4217</MeasureSchema>
            <MeasureValue>USD</MeasureValue>
            <MeasureNamespace>iso4217</MeasureNamespace>
          </StandardMeasure>
          <Scale>0</Scale>
        </Unit>
        <Unit>
          <UnitID>Pure</UnitID>
          <UnitType>Standard</UnitType>
          <StandardMeasure>
            <MeasureSchema>http://www.xbrl.org/2003/instance</MeasureSchema>
            <MeasureValue>pure</MeasureValue>
            <MeasureNamespace>xbrli</MeasureNamespace>
          </StandardMeasure>
          <Scale>0</Scale>
        </Unit>
        <Unit>
          <UnitID>USDEPS</UnitID>
          <UnitType>Divide</UnitType>
          <NumeratorMeasure>
            <MeasureSchema>http://www.xbrl.org/2003/iso4217</MeasureSchema>
            <MeasureValue>USD</MeasureValue>
            <MeasureNamespace>iso4217</MeasureNamespace>
          </NumeratorMeasure>
          <DenominatorMeasure>
            <MeasureSchema>http://www.xbrl.org/2003/instance</MeasureSchema>
            <MeasureValue>shares</MeasureValue>
            <MeasureNamespace>xbrli</MeasureNamespace>
          </DenominatorMeasure>
          <Scale>0</Scale>
        </Unit>
        <Unit>
          <UnitID>Shares</UnitID>
          <UnitType>Standard</UnitType>
          <StandardMeasure>
            <MeasureSchema>http://www.xbrl.org/2003/instance</MeasureSchema>
            <MeasureValue>shares</MeasureValue>
            <MeasureNamespace>xbrli</MeasureNamespace>
          </StandardMeasure>
          <Scale>0</Scale>
        </Unit>
      </Units>
      <CurrencySymbol>$</CurrencySymbol>
    </Column>
  </Columns>
  <Rows>
    <Row>
      <Id>2</Id>
      <Label>Organization and Basis of Presentation [Abstract]</Label>
      <Level>0</Level>
      <ElementName>us-gaap_GeneralPoliciesAbstract</ElementName>
      <ElementPrefix>us-gaap</ElementPrefix>
      <IsBaseElement>true</IsBaseElement>
      <BalanceType>na</BalanceType>
      <PeriodType>duration</PeriodType>
      <ShortDefinition>No definition available.</ShortDefinition>
      <IsReportTitle>false</IsReportTitle>
      <IsSegmentTitle>false</IsSegmentTitle>
      <IsSubReportEnd>false</IsSubReportEnd>
      <IsCalendarTitle>false</IsCalendarTitle>
      <IsTuple>false</IsTuple>
      <IsAbstractGroupTitle>true</IsAbstractGroupTitle>
      <IsEquityPrevioslyReportedAsRow>false</IsEquityPrevioslyReportedAsRow>
      <IsEquityAdjustmentRow>false</IsEquityAdjustmentRow>
      <IsBeginningBalance>false</IsBeginningBalance>
      <IsEndingBalance>false</IsEndingBalance>
      <IsReverseSign>false</IsReverseSign>
      <PreferredLabelRole />
      <IsEPS>false</IsEPS>
      <FootnoteIndexer />
      <Cells>
        <Cell>
          <Id>1</Id>
          <ShowCurrencySymbol>false</ShowCurrencySymbol>
          <IsNumeric>false</IsNumeric>
          <IsRatio>false</IsRatio>
          <DisplayZeroAsNone>false</DisplayZeroAsNone>
          <NumericAmount>0</NumericAmount>
          <RoundedNumericAmount>0</RoundedNumericAmount>
          <NonNumbericText />
          <NonNumericTextHeader />
          <FootnoteIndexer />
          <hasSegments>false</hasSegments>
          <hasScenarios>false</hasScenarios>
          <DisplayDateInUSFormat>false</DisplayDateInUSFormat>
        </Cell>
      </Cells>
      <OriginalInstanceReportColumns />
      <ElementDataType>xbrli:stringItemType</ElementDataType>
      <SimpleDataType>string</SimpleDataType>
      <ElementDefenition>No definition available.</ElementDefenition>
      <IsTotalLabel>false</IsTotalLabel>
    </Row>
    <Row>
      <Id>3</Id>
      <Label>Organization and Basis of Presentation</Label>
      <Level>1</Level>
      <ElementName>us-gaap_OrganizationConsolidationAndPresentationOfFinancialStatementsDisclosureTextBlock</ElementName>
      <ElementPrefix>us-gaap</ElementPrefix>
      <IsBaseElement>true</IsBaseElement>
      <BalanceType>na</BalanceType>
      <PeriodType>duration</PeriodType>
      <ShortDefinition>No definition available.</ShortDefinition>
      <IsReportTitle>false</IsReportTitle>
      <IsSegmentTitle>false</IsSegmentTitle>
      <IsSubReportEnd>false</IsSubReportEnd>
      <IsCalendarTitle>false</IsCalendarTitle>
      <IsTuple>false</IsTuple>
      <IsAbstractGroupTitle>false</IsAbstractGroupTitle>
      <IsEquityPrevioslyReportedAsRow>false</IsEquityPrevioslyReportedAsRow>
      <IsEquityAdjustmentRow>false</IsEquityAdjustmentRow>
      <IsBeginningBalance>false</IsBeginningBalance>
      <IsEndingBalance>false</IsEndingBalance>
      <IsReverseSign>false</IsReverseSign>
      <PreferredLabelRole>verboselabel</PreferredLabelRole>
      <IsEPS>false</IsEPS>
      <FootnoteIndexer />
      <Cells>
        <Cell>
          <Id>1</Id>
          <ShowCurrencySymbol>false</ShowCurrencySymbol>
          <IsNumeric>false</IsNumeric>
          <IsRatio>false</IsRatio>
          <DisplayZeroAsNone>false</DisplayZeroAsNone>
          <NumericAmount>0</NumericAmount>
          <RoundedNumericAmount>0</RoundedNumericAmount>
          <NonNumbericText>&lt;!--DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd" --&gt;
   &lt;!-- Begin Block Tagged Note 1 - us-gaap:OrganizationConsolidationAndPresentationOfFinancialStatementsDisclosureTextBlock--&gt;
   &lt;div align="left" style="font-family: 'Times New Roman',Times,serif"&gt;
   &lt;!-- xbrl,ns --&gt;
   &lt;!-- xbrl,nx --&gt;
   &lt;div align="center" style="font-size: 10pt; margin-top: 0pt"&gt;&lt;b&gt;
   &lt;/b&gt;
   &lt;/div&gt;
   &lt;div align="left" style="font-size: 10pt; margin-top: 12pt"&gt;&lt;b&gt;Note 1&amp;#8212;Organization and Basis of Presentation&lt;/b&gt;
   &lt;/div&gt;
   &lt;div align="left" style="font-size: 10pt; margin-top: 6pt"&gt;&lt;b&gt;Organization&lt;/b&gt;
   &lt;/div&gt;
   &lt;div align="left" style="font-size: 10pt; margin-top: 6pt"&gt;NYSE Euronext is a holding company that, through its subsidiaries, operates the following
   securities exchanges: the New York Stock Exchange (&amp;#8220;NYSE&amp;#8221;), NYSE Arca, Inc. (&amp;#8220;NYSE Arca&amp;#8221;) and NYSE
   Amex LLC (&amp;#8220;NYSE Amex&amp;#8221;) in the United States and the five European-based exchanges that comprise
   Euronext N.V. (&amp;#8220;Euronext&amp;#8221;)&amp;#8212;the Paris, Amsterdam, Brussels and Lisbon stock exchanges, as well as
   derivatives markets in London, Paris, Amsterdam, Brussels and Lisbon (collectively, &amp;#8220;NYSE Liffe&amp;#8221;)
   and the United States futures market, NYSE Liffe US, LLC (&amp;#8220;NYSE Liffe US&amp;#8221;). NYSE Euronext is a
   global provider of securities listing, trading, market data products, and software and technology
   services. NYSE Euronext was formed in connection with the April&amp;#160;4, 2007 combination of NYSE Group
   (which was formed in connection with the March&amp;#160;7, 2006 merger of the NYSE and Archipelago) and
   Euronext. NYSE Euronext common stock is dually listed on the NYSE and Euronext Paris under the
   symbol &amp;#8220;NYX.&amp;#8221;
   &lt;/div&gt;
   &lt;div align="left" style="font-size: 10pt; margin-top: 12pt"&gt;&lt;b&gt;Basis of Presentation&lt;/b&gt;
   &lt;/div&gt;
   &lt;div align="left" style="font-size: 10pt; margin-top: 6pt"&gt;The accompanying condensed unaudited consolidated financial statements include the accounts of NYSE
   Euronext and its subsidiaries.
   &lt;/div&gt;
   &lt;div align="left" style="font-size: 10pt; margin-top: 6pt"&gt;The accompanying condensed unaudited consolidated financial statements are prepared in accordance
   with accounting principles generally accepted in the U.S. and reflect all adjustments, consisting
   of only normal recurring adjustments, that are, in the opinion of management, necessary for a fair
   statement of the results for the period. All material intercompany accounts and transactions have
   been eliminated in consolidation. Certain information and footnote disclosures normally required in
   financial statements under accounting principles generally accepted in the U.S. have been condensed
   or omitted; however, management believes that the disclosures are adequate to make the information
   presented not misleading.
   &lt;/div&gt;
   &lt;div align="left" style="font-size: 10pt; margin-top: 6pt"&gt;The preparation of these condensed unaudited consolidated financial statements, in conformity with
   accounting principles generally accepted in the U.S., requires management to make estimates and
   assumptions that affect the reported amounts of assets and liabilities as of the date of the
   financial statements and the reported amounts of revenues and expenses during the period. Actual
   results could be materially different from these estimates. Certain prior period amounts have been
   reclassified to conform to the current period&amp;#8217;s presentation.
   &lt;/div&gt;
   &lt;div align="left" style="font-size: 10pt; margin-top: 6pt"&gt;We revised our reportable business segments effective in the first quarter of 2010. The new
   segments are Derivatives, Cash Trading and Listings, and Information Services and Technology
   Solutions. Historical financial results have been revised to reflect this change.
   &lt;/div&gt;
   &lt;div align="left" style="font-size: 10pt; margin-top: 6pt"&gt;The condensed consolidated financial statements are unaudited and should be read in conjunction
   with the audited financial statements of NYSE Euronext as of and for the year ended December&amp;#160;31,
   2009. Operating results for the three and six months ended June&amp;#160;30, 2010 are not necessarily
   indicative of the results that may be expected for the year ending December&amp;#160;31, 2010.
   &lt;/div&gt;
   &lt;/div&gt;
</NonNumbericText>
          <NonNumericTextHeader>&lt;!--DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd" --&gt;
   &lt;!-- Begin Block Tagged Note</NonNumericTextHeader>
          <FootnoteIndexer />
          <hasSegments>false</hasSegments>
          <hasScenarios>false</hasScenarios>
          <DisplayDateInUSFormat>false</DisplayDateInUSFormat>
        </Cell>
      </Cells>
      <OriginalInstanceReportColumns />
      <ElementDataType>us-types:textBlockItemType</ElementDataType>
      <SimpleDataType>textblock</SimpleDataType>
      <ElementDefenition>Description containing the entire organization, consolidation and basis of presentation of financial statements disclosure. May be provided in more than one note to the financial statements, as long as users are provided with an understanding of (1) the significant judgments and assumptions made by an enterprise in determining whether it must consolidate a VIE and/or disclose information about its involvement with a VIE, (2) the nature of restrictions on a consolidated VIE's assets reported by an enterprise in its statement of financial position, including the carrying amounts of such assets, (3) the nature of, and changes in, the risks associated with an enterprise's involvement with the VIE, and (4) how an enterprise's involvement with the VIE affects the enterprise's financial position, financial performance, and cash flows.  Describes procedure if disclosures are provided in more than one note to the financial statements.</ElementDefenition>
      <ElementReferences>Reference 1: http://www.xbrl.org/2003/role/presentationRef
 -Publisher FASB
 -Name FASB Staff Position (FSP)
 -Number FAS140-4 and FIN46(R)-8
 -Paragraph 8, C1, C7

Reference 2: http://www.xbrl.org/2003/role/presentationRef
 -Publisher AICPA
 -Name Accounting Research Bulletin (ARB)
 -Number 51
 -Paragraph 2-6

Reference 3: http://www.xbrl.org/2003/role/presentationRef
 -Publisher AICPA
 -Name Statement of Position (SOP)
 -Number 94-6
 -Paragraph 10

Reference 4: http://www.xbrl.org/2003/role/presentationRef
 -Publisher FASB
 -Name FASB Interpretation (FIN)
 -Number 46R
 -Paragraph 4, 14, 15

</ElementReferences>
      <IsTotalLabel>false</IsTotalLabel>
    </Row>
  </Rows>
  <Footnotes />
  <NumberOfCols>1</NumberOfCols>
  <NumberOfRows>2</NumberOfRows>
  <HasScenarios>false</HasScenarios>
  <MonetaryRoundingLevel>UnKnown</MonetaryRoundingLevel>
  <SharesRoundingLevel>UnKnown</SharesRoundingLevel>
  <PerShareRoundingLevel>UnKnown</PerShareRoundingLevel>
  <HasPureData>false</HasPureData>
  <SharesShouldBeRounded>true</SharesShouldBeRounded>
</InstanceReport>
