<?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>2101100 - Disclosure - Nature of Business</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>0001113481</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_OrganizationConsolidationAndPresentationOfFinancialStatementsAbstract</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>Organization, Consolidation and Presentation of Financial Statements [Abstract]</Label></Row><Row FlagID="0"><Id>2</Id><IsAbstractGroupTitle>false</IsAbstractGroupTitle><LabelSeparator>

</LabelSeparator><Level>2</Level><ElementName>us-gaap_NatureOfOperations</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="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%;text-align:left;text-indent:18px;font-size:10pt;"&gt;&lt;font style="font-family:inherit;font-size:10pt;font-weight:bold;"&gt; Nature of Business&lt;/font&gt;&lt;/div&gt;&lt;div style="line-height:120%;text-align:left;font-size:10pt;"&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt;&lt;br clear="none"/&gt;&lt;/font&gt;&lt;/div&gt;&lt;div style="line-height:120%;text-align:justify;text-indent:18px;font-size:10pt;"&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt;The Medicines Company (the Company) is a global biopharmaceutical company focused on saving lives, alleviating suffering and improving the economic efficiency of the world's leading hospitals. The Company markets Angiomax&lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt;&lt;sup style="vertical-align:top;line-height:120%;font-size:7pt"&gt;&amp;#174; &lt;/sup&gt;&lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt;(bivalirudin), Recothrom&lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt;&lt;sup style="vertical-align:top;line-height:120%;font-size:7pt"&gt;&amp;#174; &lt;/sup&gt;&lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt;Thrombin, topical (Recombinant) and&lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt;&lt;sup style="vertical-align:top;line-height:120%;font-size:7pt"&gt; &lt;/sup&gt;&lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt;Cleviprex&lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt;&lt;sup style="vertical-align:top;line-height:120%;font-size:7pt"&gt;&amp;#174;&lt;/sup&gt;&lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt; (clevidipine) injectable emulsion.  The Company also has a pipeline of acute and intensive care hospital products in development, including &lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;color:#000000;text-decoration:none;"&gt;three&lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt; late-stage development product candidates, cangrelor, oritavancin and IONSYS&lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt;&lt;sup style="vertical-align:top;line-height:120%;font-size:7pt"&gt;TM&lt;/sup&gt;&lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt; (fentanyl iontophoretic transdermal system) and early stage development product candidates MDCO-216 and ALN-PCS02 and ALN-PCSsc of its ALN-PCS program.  The Company believes that these marketed products and products in development possess favorable attributes that competitive products do not provide, can satisfy unmet medical needs in the acute and intensive care hospital product market and offer, or, in the case of its products in development, have the potential to offer, improved performance to hospital businesses.  &lt;/font&gt;&lt;/div&gt;&lt;div style="line-height:120%;text-align:justify;text-indent:18px;font-size:10pt;"&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt;&lt;br clear="none"/&gt;&lt;/font&gt;&lt;/div&gt;&lt;div style="line-height:120%;text-align:justify;text-indent:18px;font-size:10pt;"&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt;In addition to these products and product candidates, the Company sells a ready-to-use formulation of Argatroban and has a portfolio of &lt;/font&gt;&lt;font style="font-family:Times New Roman;font-size:10pt;color:#000000;font-style:normal;font-weight:normal;text-decoration:none;"&gt;ten&lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt; generic drugs, which the Company refers to as its acute care generic products, that the Company has the non-exclusive right to market in the United States. The Company began selling &lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;color:#000000;text-decoration:none;"&gt;three&lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt; of its acute care generic products, midazolam, ondansetron and rocuronium, in the first quarter of 2013.  The Company also co-promotes the oral tablet antiplatelet medicine&lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;color:#00c000;"&gt; &lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt;BRILINTA&lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt;&lt;sup style="vertical-align:top;line-height:120%;font-size:7pt"&gt;&amp;#174;&lt;/sup&gt;&lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt; (ticagrelor) in the United States, as part of its global collaboration agreement with AstraZeneca LP (AstraZeneca).   &lt;/font&gt;&lt;/div&gt;&lt;div style="line-height:120%;text-align:justify;text-indent:18px;font-size:10pt;"&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt;&lt;br clear="none"/&gt;&lt;/font&gt;&lt;/div&gt;&lt;div style="line-height:120%;text-align:justify;text-indent:18px;font-size:10pt;"&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt;On August 5, 2013, the Company completed its acquisition of all of the outstanding equity of ProFibrix B.V. (ProFibrix) pursuant to a share purchase agreement entered into with ProFibrix and its equityholders on June 4, 2013.  Upon the completion of the acquisition, ProFibrix became a wholly owned subsidiary of the Company.  Under the share purchase agreement, the closing of the transaction was subject to the Company's satisfactory review of the then pending Phase 3 clinical trial results of ProFibrix's lead biologic, Fibrocaps&amp;#8482;.  Fibrocaps is a dry powder topical formulation of fibrinogen and thrombin being developed to help stop bleeding during surgery.  Following the Company's review of the Phase 3 trial results, on August 2, 2013, the Company notified ProFibrix that it wished to proceed with the consummation of the transaction.&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>The entire disclosure for the nature of an entity's business, the major products or services it sells or provides and its principal markets, including the locations of those markets. If the entity operates in more than one business, the disclosure also indicates the relative importance of its operations in each business and the basis for the determination (for example, assets, revenues, or earnings).</ElementDefenition><ElementReferences>Reference 1: http://www.xbrl.org/2003/role/presentationRef

 -Publisher FASB

 -Name Accounting Standards Codification

 -Topic 275

 -SubTopic 10

 -Section 50

 -Paragraph 2

 -URI http://asc.fasb.org/extlink&amp;oid=6927468&amp;loc=d3e6003-108592



</ElementReferences><IsTotalLabel>false</IsTotalLabel><UnitID>0</UnitID><Label>Nature of Business</Label></Row></Rows><Footnotes /><IsEquityReport>false</IsEquityReport><ReportName>Nature of Business</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.themedicinescompany.com/role/NatureOfBusiness</RoleURI><NumberOfCols>1</NumberOfCols><NumberOfRows>2</NumberOfRows></InstanceReport>
