<?xml version="1.0"?>
<ownershipDocument>

    <schemaVersion>X0306</schemaVersion>

    <documentType>4</documentType>

    <periodOfReport>2022-01-24</periodOfReport>

    <notSubjectToSection16>1</notSubjectToSection16>

    <issuer>
        <issuerCik>0000036270</issuerCik>
        <issuerName>M&amp;T BANK CORP</issuerName>
        <issuerTradingSymbol>MTB</issuerTradingSymbol>
    </issuer>

    <reportingOwner>
        <reportingOwnerId>
            <rptOwnerCik>0001318534</rptOwnerCik>
            <rptOwnerName>Farrell William J II</rptOwnerName>
        </reportingOwnerId>
        <reportingOwnerAddress>
            <rptOwnerStreet1>1100 NORTH MARKET STREET</rptOwnerStreet1>
            <rptOwnerStreet2>RODNEY SQUARE NO.</rptOwnerStreet2>
            <rptOwnerCity>WILMINGTON</rptOwnerCity>
            <rptOwnerState>DE</rptOwnerState>
            <rptOwnerZipCode>19890</rptOwnerZipCode>
            <rptOwnerStateDescription></rptOwnerStateDescription>
        </reportingOwnerAddress>
        <reportingOwnerRelationship>
            <isDirector>0</isDirector>
            <isOfficer>1</isOfficer>
            <isTenPercentOwner>0</isTenPercentOwner>
            <isOther>0</isOther>
            <officerTitle>Executive Vice President</officerTitle>
        </reportingOwnerRelationship>
    </reportingOwner>

    <nonDerivativeTable>
        <nonDerivativeHolding>
            <securityTitle>
                <value>Common Stock</value>
            </securityTitle>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>13348.86</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </nonDerivativeHolding>
    </nonDerivativeTable>

    <footnotes></footnotes>

    <remarks>This filing is being made to reflect the termination of the reporting person's status as an insider of M&amp;T Bank Corporation effective January 24, 2022.  Unless otherwise indicated, the reporting person's holdings are as of such date.</remarks>

    <ownerSignature>
        <signatureName>By: Stephen T. Wilson, Esq.
(Attorney-In-Fact)</signatureName>
        <signatureDate>2022-02-01</signatureDate>
    </ownerSignature>
</ownershipDocument>
