<?xml version="1.0"?><ownershipDocument><schemaVersion>X0201</schemaVersion>    <documentType>4</documentType><periodOfReport>2003-10-30</periodOfReport><notSubjectToSection16>0</notSubjectToSection16><issuer><issuerCik>0000895930</issuerCik><issuerName>AMSURG CORP</issuerName>        <issuerTradingSymbol>amsg</issuerTradingSymbol></issuer><reportingOwner><reportingOwnerId><rptOwnerCik>0001203385</rptOwnerCik><rptOwnerName>GUTHRIE DEBORAH A</rptOwnerName></reportingOwnerId><reportingOwnerAddress><rptOwnerStreet1>2213 WASHINGTON CIRCLE DRIVE</rptOwnerStreet1><rptOwnerStreet2></rptOwnerStreet2><rptOwnerCity>WASHINGTON</rptOwnerCity><rptOwnerState>DC</rptOwnerState><rptOwnerZipCode>20007</rptOwnerZipCode><rptOwnerStateDescription></rptOwnerStateDescription></reportingOwnerAddress><reportingOwnerRelationship><isDirector>1</isDirector><isOfficer>0</isOfficer><isTenPercentOwner>0</isTenPercentOwner><isOther>0</isOther></reportingOwnerRelationship></reportingOwner><nonDerivativeTable><nonDerivativeTransaction><securityTitle><value>Common stock</value></securityTitle><transactionDate><value>2003-10-30</value></transactionDate><deemedExecutionDate></deemedExecutionDate><transactionCoding><transactionFormType>4</transactionFormType><transactionCode>S</transactionCode><equitySwapInvolved>0</equitySwapInvolved></transactionCoding><transactionTimeliness><value></value></transactionTimeliness><transactionAmounts><transactionShares><value>2000</value></transactionShares><transactionPricePerShare><value>35</value></transactionPricePerShare><transactionAcquiredDisposedCode><value>D</value></transactionAcquiredDisposedCode></transactionAmounts><postTransactionAmounts><sharesOwnedFollowingTransaction><value>5966</value></sharesOwnedFollowingTransaction></postTransactionAmounts><ownershipNature><directOrIndirectOwnership><value>D</value></directOrIndirectOwnership></ownershipNature></nonDerivativeTransaction><nonDerivativeHolding><securityTitle><value>Common stock</value></securityTitle><postTransactionAmounts><sharesOwnedFollowingTransaction><value>500</value></sharesOwnedFollowingTransaction></postTransactionAmounts><ownershipNature><directOrIndirectOwnership><value>I</value></directOrIndirectOwnership><natureOfOwnership><value>Shares held by Capitol Health Consultants, Inc., which Ms. Guthrie is President and CEO</value></natureOfOwnership></ownershipNature></nonDerivativeHolding></nonDerivativeTable><footnotes></footnotes><remarks></remarks><ownerSignature><signatureName>/s/ Debora A. Guthrie</signatureName><signatureDate>2003-10-30</signatureDate></ownerSignature></ownershipDocument>

