﻿<?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>Commitments and Contingencies</ReportName>
  <ReportLongName>0209 - Disclosure - Commitments and Contingencies</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>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>Commitments and Contingencies [Abstract]</Label>
      <Level>0</Level>
      <ElementName>me_CommitmentsAndContingenciesAbstract</ElementName>
      <ElementPrefix>me</ElementPrefix>
      <IsBaseElement>false</IsBaseElement>
      <BalanceType>na</BalanceType>
      <PeriodType>duration</PeriodType>
      <ShortDefinition>Commitments and Contingencies.</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>Commitments and Contingencies.</ElementDefenition>
      <IsTotalLabel>false</IsTotalLabel>
    </Row>
    <Row>
      <Id>3</Id>
      <Label>Commitments and Contingencies</Label>
      <Level>1</Level>
      <ElementName>us-gaap_CommitmentsAndContingenciesDisclosureTextBlock</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 9 - us-gaap:CommitmentsAndContingenciesDisclosureTextBlock--&gt;
   &lt;div style="font-family: 'Times New Roman',Times,serif"&gt;
   &lt;div align="left" style="font-size: 10pt; margin-top: 12pt"&gt;&lt;b&gt;9. Commitments and Contingencies&lt;/b&gt;
   &lt;/div&gt;
   &lt;div align="left" style="font-size: 10pt; margin-top: 6pt"&gt;&amp;#160;&amp;#160;&amp;#160;&amp;#160;&amp;#160;&lt;i&gt;Minimum Future Lease Payments &amp;#8212; &lt;/i&gt;The Company leases certain office facilities and other
   equipment under long-term operating lease arrangements. Minimum future lease obligations under the
   Company&amp;#8217;s operating leases in effect at June&amp;#160;30, 2010 are as follows:
   &lt;/div&gt;
   &lt;div align="center"&gt;
   &lt;table style="font-size: 10pt; text-align: left" cellspacing="0" border="0" cellpadding="0" width="100%"&gt;
   &lt;!-- Begin Table Head --&gt;
   &lt;tr valign="bottom"&gt;
       &lt;td width="88%"&gt;&amp;#160;&lt;/td&gt;
       &lt;td width="5%"&gt;&amp;#160;&lt;/td&gt;
       &lt;td width="3%"&gt;&amp;#160;&lt;/td&gt;
       &lt;td width="1%"&gt;&amp;#160;&lt;/td&gt;
       &lt;td width="3%"&gt;&amp;#160;&lt;/td&gt;
   &lt;/tr&gt;
   &lt;tr style="font-size: 8pt" valign="bottom"&gt;
       &lt;td&gt;&amp;#160;&lt;/td&gt;
       &lt;td&gt;&amp;#160;&lt;/td&gt;
       &lt;td nowrap="nowrap" align="center" colspan="3" style="border-bottom: 1px solid #000000"&gt;&lt;b&gt;(In thousands)&lt;/b&gt;&lt;/td&gt;
   &lt;/tr&gt;
   &lt;!-- End Table Head --&gt;
   &lt;!-- Begin Table Body --&gt;
   &lt;tr valign="bottom" style="background: #cceeff"&gt;
       &lt;td&gt;
   &lt;div style="margin-left:15px; text-indent:-15px"&gt;2011
   &lt;/div&gt;&lt;/td&gt;
       &lt;td&gt;&amp;#160;&lt;/td&gt;
       &lt;td align="right"&gt;$&lt;/td&gt;
       &lt;td align="right"&gt;1,898&lt;/td&gt;
       &lt;td&gt;&amp;#160;&lt;/td&gt;
   &lt;/tr&gt;
   &lt;tr valign="bottom"&gt;
       &lt;td&gt;
   &lt;div style="margin-left:15px; text-indent:-15px"&gt;2012
   &lt;/div&gt;&lt;/td&gt;
       &lt;td&gt;&amp;#160;&lt;/td&gt;
       &lt;td&gt;&amp;#160;&lt;/td&gt;
       &lt;td align="right"&gt;3,695&lt;/td&gt;
       &lt;td&gt;&amp;#160;&lt;/td&gt;
   &lt;/tr&gt;
   &lt;tr valign="bottom" style="background: #cceeff"&gt;
       &lt;td&gt;
   &lt;div style="margin-left:15px; text-indent:-15px"&gt;2013
   &lt;/div&gt;&lt;/td&gt;
       &lt;td&gt;&amp;#160;&lt;/td&gt;
       &lt;td&gt;&amp;#160;&lt;/td&gt;
       &lt;td align="right"&gt;3,435&lt;/td&gt;
       &lt;td&gt;&amp;#160;&lt;/td&gt;
   &lt;/tr&gt;
   &lt;tr valign="bottom"&gt;
       &lt;td&gt;
   &lt;div style="margin-left:15px; text-indent:-15px"&gt;2014
   &lt;/div&gt;&lt;/td&gt;
       &lt;td&gt;&amp;#160;&lt;/td&gt;
       &lt;td&gt;&amp;#160;&lt;/td&gt;
       &lt;td align="right"&gt;3,308&lt;/td&gt;
       &lt;td&gt;&amp;#160;&lt;/td&gt;
   &lt;/tr&gt;
   &lt;tr valign="bottom" style="background: #cceeff"&gt;
       &lt;td&gt;
   &lt;div style="margin-left:15px; text-indent:-15px"&gt;2015 and thereafter
   &lt;/div&gt;&lt;/td&gt;
       &lt;td&gt;&amp;#160;&lt;/td&gt;
       &lt;td&gt;&amp;#160;&lt;/td&gt;
       &lt;td align="right"&gt;13,065&lt;/td&gt;
       &lt;td&gt;&amp;#160;&lt;/td&gt;
   &lt;/tr&gt;
   &lt;!-- End Table Body --&gt;
   &lt;/table&gt;
   &lt;/div&gt;
   &lt;div align="left" style="font-size: 10pt; margin-top: 6pt"&gt;&amp;#160;&amp;#160;&amp;#160;&amp;#160;&amp;#160;&lt;i&gt;Other Commitments &amp;#8212; &lt;/i&gt;In the ordinary course of business, the Company enters into long-term
   commitments to purchase seismic data and other geological information such as maps, logs and
   studies. The minimum annual payments under these contracts are $4.8&amp;#160;million in 2011.
   &lt;/div&gt;
   &lt;div align="left" style="font-size: 10pt; margin-top: 6pt"&gt;&amp;#160;&amp;#160;&amp;#160;&amp;#160;&amp;#160;&lt;i&gt;Insurance Matters&lt;/i&gt;
   &lt;/div&gt;
   &lt;div align="left" style="font-size: 10pt; margin-top: 6pt"&gt;&amp;#160;&amp;#160;&amp;#160;&amp;#160;&amp;#160;&lt;i&gt;Current Insurance Against Hurricanes&lt;/i&gt;
   Mariner is a member of OIL Insurance Limited (&amp;#8220;OIL&amp;#8221;), an energy industry insurance
   cooperative, which provides Mariner windstorm insurance coverage. During 2009, the coverage was
   subject to a $10.0&amp;#160;million per-occurrence deductible, a $250.0&amp;#160;million per-occurrence loss limit,
   and a $750.0&amp;#160;million industry aggregate per-event loss limit. Effective January&amp;#160;1, 2010, the
   coverage is subject to a $10.0&amp;#160;million per-occurrence deductible; a $150.0&amp;#160;million per-occurrence
   loss limit per member that Mariner elected to supplement with $25.0&amp;#160;million in additional coverage
   which if used, would be repayable, interest free, over five years; an annual maximum of $300.0
   million per member; and a $750.0&amp;#160;million industry aggregate per-event loss limit. Annual industry
   windstorm losses of $300.0&amp;#160;million or less will be mutualized among all members. Annual industry
   windstorm losses exceeding $300.0&amp;#160;million will be mutualized among windstorm members in two pools,
   one for offshore and one for onshore, with future premiums based upon a pool&amp;#8217;s loss experience and
   a member&amp;#8217;s weighted percent of the pool&amp;#8217;s asset base. Mariner anticipates these changes to increase
   its loss retention by approximately $100.0&amp;#160;million for windstorm losses, which it expects to either
   self insure, insure through the commercial market, insure through the purchase of additional OIL
   coverage or a combination of these.
   &lt;/div&gt;
   &lt;div align="left" style="font-size: 10pt; margin-top: 6pt"&gt;&amp;#160;&amp;#160;&amp;#160;&amp;#160;&amp;#160;Mariner annually considers whether the commercial market offers supplemental or excess
   insurance that would, based on Mariner&amp;#8217;s historical experience, supplement its OIL coverage on a
   cost-effective basis. In 2010, Mariner elected to purchase insurance from the commercial market to
   supplement the reduced windstorm coverage offered by OIL. The supplemental insurance will provide
   up to an additional $78.3&amp;#160;million of aggregate annual coverage in respect of windstorms, of which
   up to $49.1&amp;#160;million could cover revenues lost as a result of constructive total losses of
   third-party owned structures through which a material amount of Mariner production is routed and
   cannot be rerouted.
   &lt;/div&gt;
   &lt;div align="left" style="font-size: 10pt; margin-top: 6pt"&gt;&amp;#160;&amp;#160;&amp;#160;&amp;#160;&amp;#160;As of June&amp;#160;30, 2010, Mariner accrued approximately $41.2&amp;#160;million for an OIL withdrawal premium
   contingency. As part of its OIL membership, Mariner is obligated to pay a withdrawal premium if it
   elects to withdraw from OIL. Mariner does not anticipate withdrawing from OIL; however, due to the
   contingency, Mariner periodically reassesses the sufficiency of its accrued withdrawal premium
   based on OIL&amp;#8217;s periodic calculation of the potential withdrawal premium in light of past losses,
   and Mariner may adjust its accrual accordingly in the future. OIL requires smaller members to
   provide a letter of credit or other acceptable security in favor of OIL to secure payment of the
   withdrawal premium. Acceptable security has included a letter of credit or a security agreement
   pursuant to which a member grants OIL a security interest in certain claim proceeds payable by OIL
   to the member. Mariner has entered into such a security agreement, granting to OIL a senior
   security interest in up to the next $50.0&amp;#160;million in excess of $100.0&amp;#160;million of Mariner&amp;#8217;s
   Hurricane Ike claim proceeds payable by OIL. Mariner has the ability to replace the security
   agreement with a letter of credit or other acceptable security in favor of OIL.
   &lt;/div&gt;
   &lt;div align="left" style="font-size: 10pt; margin-top: 6pt"&gt;&amp;#160;&amp;#160;&amp;#160;&amp;#160;&amp;#160;&lt;i&gt;Hurricane Ike (2008)&lt;/i&gt;
   &lt;/div&gt;
   &lt;div align="left" style="font-size: 10pt; margin-top: 6pt"&gt;&amp;#160;&amp;#160;&amp;#160;&amp;#160;&amp;#160;In 2008, the Company&amp;#8217;s operations were adversely affected by Hurricane Ike. The hurricane
   resulted in shut-in and delayed production as well as facility repairs and replacement expenses.
   The Company estimates that repairs and plugging and abandonment costs resulting from Hurricane Ike
   will total approximately $160.0&amp;#160;million net to Mariner&amp;#8217;s interest. OIL has advised the Company that
   industry-wide damages from Hurricane Ike are expected to substantially exceed OIL&amp;#8217;s $750.0&amp;#160;million
   industry aggregate per event loss limit and that OIL expects to initially prorate the payout of all
   OIL members&amp;#8217; Hurricane Ike claims at approximately 50%, subject to further adjustment. OIL also has
   indicated that the scaling factor it expects to apply to Mariner&amp;#8217;s Hurricane Ike claims will result
   in settlement at less than 70%. Mariner expects that approximately 75% of the shortfall in its
   primary insurance coverage will be covered under applicable commercial excess coverage. In respect
   of Hurricane Ike claims that the Company made through June&amp;#160;30, 2010, the Company received
   approximately $37.0&amp;#160;million from OIL and $14.0&amp;#160;million from excess carriers. Although in 2009
   Mariner started receiving payment in respect of its Hurricane Ike claims, due to the magnitude of
   the storm and the complexity of the insurance claims being processed by the insurance industry,
   Mariner expects to maintain a potentially significant insurance receivable through 2010 while it
   actively pursues settlement.
   &lt;/div&gt;
   &lt;div align="left" style="font-size: 10pt; margin-top: 6pt"&gt;&amp;#160;&amp;#160;&amp;#160;&amp;#160;&amp;#160;&lt;i&gt;Litigation &amp;#8212; &lt;/i&gt;The Company, in the ordinary course of business, is a claimant and/or a defendant
   in various legal proceedings, including proceedings as to which the Company has insurance coverage
   and those that may involve the filing of liens against the Company or its assets. The Company does
   not consider its exposure in these proceedings, individually or in the aggregate, to be material.
   &lt;/div&gt;
   &lt;!-- Folio --&gt;
   &lt;!-- /Folio --&gt;
   &lt;/div&gt;
   &lt;!-- PAGEBREAK --&gt;
   &lt;div style="font-family: 'Times New Roman',Times,serif"&gt;
   &lt;div align="left" style="font-size: 10pt; margin-top: 6pt"&gt;&amp;#160;&amp;#160;&amp;#160;&amp;#160;&amp;#160;&lt;i&gt;Letters of Credit &lt;/i&gt;&amp;#8212; Mariner&amp;#8217;s bank credit facility has a letter of credit subfacility of up to
   $50.0&amp;#160;million that is included as a use of the borrowing base. As of June&amp;#160;30, 2010, four such
   letters of credit totaling $4.7&amp;#160;million were outstanding of which $4.2&amp;#160;million is required for
   plugging and abandonment obligations at certain of Mariner&amp;#8217;s offshore fields.
   &lt;/div&gt;
   &lt;div align="left" style="font-size: 10pt; margin-top: 6pt"&gt;&amp;#160;&amp;#160;&amp;#160;&amp;#160;&amp;#160;&lt;i&gt;Gulf of Mexico Oil Spill &lt;/i&gt;&amp;#8212; As a result of the &lt;i&gt;Deepwater Horizon &lt;/i&gt;incidents in April&amp;#160;2010, the
   U.S. Department of Interior (DOI)&amp;#160;has issued a series of reforms to the oversight and management of
   offshore drilling activities on the federal Outer Continental Shelf (OCS). On July&amp;#160;12, 2010, the
   Secretary of the DOI directed the Bureau of Ocean Energy Management, Regulation and Enforcement, to
   issue a suspension until November&amp;#160;30, 2010 of drilling activities that use subsea blowout
   preventers or surface blowout preventers on floating facilities. Mariner&amp;#8217;s Gulf of Mexico offshore
   operations have been impacted and likely may be impacted in the future by increased regulatory
   oversight, which may increase the cost of OCS wells, such as Lucius, Heidelberg and Bass Lite, and
   delay drilling and production therefrom.
   &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>Includes disclosure of commitments and contingencies. This element may be used as a single block of text to encapsulate the entire disclosure including data and tables.</ElementDefenition>
      <ElementReferences>Reference 1: http://www.xbrl.org/2003/role/presentationRef
 -Publisher FASB
 -Name FASB Interpretation (FIN)
 -Number 14
 -Paragraph 3

Reference 2: http://www.xbrl.org/2003/role/presentationRef
 -Publisher FASB
 -Name Statement of Financial Accounting Standard (FAS)
 -Number 5
 -Paragraph 9, 10, 11, 12

</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>
