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

    <schemaVersion>X0306</schemaVersion>

    <documentType>4</documentType>

    <periodOfReport>2019-06-17</periodOfReport>

    <notSubjectToSection16>0</notSubjectToSection16>

    <issuer>
        <issuerCik>0001214935</issuerCik>
        <issuerName>AllianzGI Convertible &amp; Income Fund</issuerName>
        <issuerTradingSymbol>NCV</issuerTradingSymbol>
    </issuer>

    <reportingOwner>
        <reportingOwnerId>
            <rptOwnerCik>0001610520</rptOwnerCik>
            <rptOwnerName>UBS Group AG</rptOwnerName>
        </reportingOwnerId>
        <reportingOwnerAddress>
            <rptOwnerStreet1>BAHNHOFSTRASSE 45</rptOwnerStreet1>
            <rptOwnerStreet2>PO BOX CH-8098</rptOwnerStreet2>
            <rptOwnerCity>ZURICH</rptOwnerCity>
            <rptOwnerState>V8</rptOwnerState>
            <rptOwnerZipCode></rptOwnerZipCode>
            <rptOwnerStateDescription>SWITZERLAND</rptOwnerStateDescription>
        </reportingOwnerAddress>
        <reportingOwnerRelationship>
            <isDirector>0</isDirector>
            <isOfficer>0</isOfficer>
            <isTenPercentOwner>1</isTenPercentOwner>
            <isOther>0</isOther>
        </reportingOwnerRelationship>
    </reportingOwner>

    <nonDerivativeTable>
        <nonDerivativeTransaction>
            <securityTitle>
                <value>Cumulative Preferred Shares Series A</value>
            </securityTitle>
            <transactionDate>
                <value>2019-06-17</value>
            </transactionDate>
            <deemedExecutionDate></deemedExecutionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>S</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
            </transactionCoding>
            <transactionTimeliness>
                <value></value>
            </transactionTimeliness>
            <transactionAmounts>
                <transactionShares>
                    <value>29</value>
                </transactionShares>
                <transactionPricePerShare>
                    <value>25.20</value>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>D</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>59</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>I</value>
                </directOrIndirectOwnership>
                <natureOfOwnership>
                    <value>By subsidiary</value>
                    <footnoteId id="F1"/>
                </natureOfOwnership>
            </ownershipNature>
        </nonDerivativeTransaction>
    </nonDerivativeTable>

    <footnotes>
        <footnote id="F1">This Statement is filed jointly by UBS Group AG for the benefit and on behalf of UBS Securities LLC, a wholly owned subsidiary of UBS Group.</footnote>
    </footnotes>

    <remarks></remarks>

    <ownerSignature>
        <signatureName>/s/ Rollins Simmons</signatureName>
        <signatureDate>2019-06-19</signatureDate>
    </ownerSignature>

    <ownerSignature>
        <signatureName>/s/ Jennifer Sator</signatureName>
        <signatureDate>2019-06-19</signatureDate>
    </ownerSignature>
</ownershipDocument>
