<?xml version="1.0"?><ownershipDocument><schemaVersion>X0201</schemaVersion>    <documentType>3</documentType><periodOfReport>2003-10-07</periodOfReport><noSecuritiesOwned>1</noSecuritiesOwned><issuer><issuerCik>0001054721</issuerCik><issuerName>BSQUARE CORP /WA</issuerName>        <issuerTradingSymbol>BSQR</issuerTradingSymbol></issuer><reportingOwner><reportingOwnerId><rptOwnerCik>0001266377</rptOwnerCik><rptOwnerName>FOURNIER ANDRE F A</rptOwnerName></reportingOwnerId><reportingOwnerAddress><rptOwnerStreet1>3150 - 139TH AVE., SE</rptOwnerStreet1><rptOwnerStreet2>SUITE 500</rptOwnerStreet2><rptOwnerCity>BELLEVUE</rptOwnerCity><rptOwnerState>WA</rptOwnerState><rptOwnerZipCode>98005</rptOwnerZipCode><rptOwnerStateDescription></rptOwnerStateDescription></reportingOwnerAddress><reportingOwnerRelationship><isDirector>0</isDirector><isOfficer>1</isOfficer><isTenPercentOwner>0</isTenPercentOwner><isOther>0</isOther><officerTitle>Vice President, Product Dev.</officerTitle></reportingOwnerRelationship></reportingOwner><footnotes></footnotes><remarks></remarks><ownerSignature><signatureName>Andre F. A. Fournier</signatureName><signatureDate>2003-10-08</signatureDate></ownerSignature></ownershipDocument>

