<?xml version="1.0"?><?xml-stylesheet type="text/xsl" href="../4_X02.xsl"?><ownershipDocument>    <schemaVersion>X0303</schemaVersion>    <documentType>4</documentType>    <periodOfReport>2010-09-03</periodOfReport>    <notSubjectToSection16>0</notSubjectToSection16>    <issuer>        <issuerCik>0001054721</issuerCik>        <issuerName>BSQUARE CORP /WA</issuerName>        <issuerTradingSymbol>BSQR</issuerTradingSymbol>    </issuer>    <reportingOwner>        <reportingOwnerId>            <rptOwnerCik>0001499878</rptOwnerCik>            <rptOwnerName>Traynor John F.K.</rptOwnerName>        </reportingOwnerId>        <reportingOwnerAddress>            <rptOwnerStreet1>110 - 110TH AVENUE, NE</rptOwnerStreet1>            <rptOwnerStreet2>SUITE 200</rptOwnerStreet2>            <rptOwnerCity>BELLEVUE</rptOwnerCity>            <rptOwnerState>WA</rptOwnerState>            <rptOwnerZipCode>98004</rptOwnerZipCode>            <rptOwnerStateDescription></rptOwnerStateDescription>        </reportingOwnerAddress>        <reportingOwnerRelationship>            <isDirector>0</isDirector>            <isOfficer>1</isOfficer>            <isTenPercentOwner>0</isTenPercentOwner>            <isOther>0</isOther>            <officerTitle>Vice President of Products</officerTitle>        </reportingOwnerRelationship>    </reportingOwner>    <nonDerivativeTable>        <nonDerivativeTransaction>            <securityTitle>                <value>Common Stock</value>            </securityTitle>            <transactionDate>                <value>2010-09-03</value>            </transactionDate>            <deemedExecutionDate></deemedExecutionDate>            <transactionCoding>                <transactionFormType>4</transactionFormType>                <transactionCode>P</transactionCode>                <equitySwapInvolved>0</equitySwapInvolved>            </transactionCoding>            <transactionTimeliness>                <value></value>            </transactionTimeliness>            <transactionAmounts>                <transactionShares>                    <value>3900</value>                </transactionShares>                <transactionPricePerShare>                    <value>2.9896</value>                </transactionPricePerShare>                <transactionAcquiredDisposedCode>                    <value>A</value>                </transactionAcquiredDisposedCode>            </transactionAmounts>            <postTransactionAmounts>                <sharesOwnedFollowingTransaction>                    <value>4900</value>                </sharesOwnedFollowingTransaction>            </postTransactionAmounts>            <ownershipNature>                <directOrIndirectOwnership>                    <value>D</value>                </directOrIndirectOwnership>            </ownershipNature>        </nonDerivativeTransaction>        <nonDerivativeTransaction>            <securityTitle>                <value>Common Stock</value>            </securityTitle>            <transactionDate>                <value>2010-09-03</value>            </transactionDate>            <deemedExecutionDate></deemedExecutionDate>            <transactionCoding>                <transactionFormType>4</transactionFormType>                <transactionCode>P</transactionCode>                <equitySwapInvolved>0</equitySwapInvolved>            </transactionCoding>            <transactionTimeliness>                <value></value>            </transactionTimeliness>            <transactionAmounts>                <transactionShares>                    <value>92</value>                </transactionShares>                <transactionPricePerShare>                    <value>2.9911</value>                </transactionPricePerShare>                <transactionAcquiredDisposedCode>                    <value>A</value>                </transactionAcquiredDisposedCode>            </transactionAmounts>            <postTransactionAmounts>                <sharesOwnedFollowingTransaction>                    <value>4992</value>                </sharesOwnedFollowingTransaction>            </postTransactionAmounts>            <ownershipNature>                <directOrIndirectOwnership>                    <value>D</value>                </directOrIndirectOwnership>            </ownershipNature>        </nonDerivativeTransaction>        <nonDerivativeTransaction>            <securityTitle>                <value>Common Stock</value>            </securityTitle>            <transactionDate>                <value>2010-09-03</value>            </transactionDate>            <deemedExecutionDate></deemedExecutionDate>            <transactionCoding>                <transactionFormType>4</transactionFormType>                <transactionCode>P</transactionCode>                <equitySwapInvolved>0</equitySwapInvolved>            </transactionCoding>            <transactionTimeliness>                <value></value>            </transactionTimeliness>            <transactionAmounts>                <transactionShares>                    <value>1008</value>                </transactionShares>                <transactionPricePerShare>                    <value>3.00</value>                </transactionPricePerShare>                <transactionAcquiredDisposedCode>                    <value>A</value>                </transactionAcquiredDisposedCode>            </transactionAmounts>            <postTransactionAmounts>                <sharesOwnedFollowingTransaction>                    <value>6000</value>                </sharesOwnedFollowingTransaction>            </postTransactionAmounts>            <ownershipNature>                <directOrIndirectOwnership>                    <value>D</value>                </directOrIndirectOwnership>            </ownershipNature>        </nonDerivativeTransaction>    </nonDerivativeTable>    <footnotes></footnotes>    <remarks></remarks>    <ownerSignature>        <signatureName>/s/ John F.K. Traynor</signatureName>        <signatureDate>2010-09-07</signatureDate>    </ownerSignature></ownershipDocument>
