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

    <schemaVersion>X0206</schemaVersion>

    <documentType>3</documentType>

    <periodOfReport>2016-01-01</periodOfReport>

    <noSecuritiesOwned>1</noSecuritiesOwned>

    <issuer>
        <issuerCik>0001636289</issuerCik>
        <issuerName>AllianzGI Diversified Income &amp; Convertible Fund</issuerName>
        <issuerTradingSymbol>ACV</issuerTradingSymbol>
    </issuer>

    <reportingOwner>
        <reportingOwnerId>
            <rptOwnerCik>0001663195</rptOwnerCik>
            <rptOwnerName>Eu Douglas</rptOwnerName>
        </reportingOwnerId>
        <reportingOwnerAddress>
            <rptOwnerStreet1>1633 BROADWAY</rptOwnerStreet1>
            <rptOwnerStreet2></rptOwnerStreet2>
            <rptOwnerCity>NEW YORK</rptOwnerCity>
            <rptOwnerState>NY</rptOwnerState>
            <rptOwnerZipCode>10009</rptOwnerZipCode>
            <rptOwnerStateDescription></rptOwnerStateDescription>
        </reportingOwnerAddress>
        <reportingOwnerRelationship>
            <isDirector>0</isDirector>
            <isOfficer>0</isOfficer>
            <isTenPercentOwner>0</isTenPercentOwner>
            <isOther>1</isOther>
            <otherText>SEE REMARKS</otherText>
        </reportingOwnerRelationship>
    </reportingOwner>

    <footnotes></footnotes>

    <remarks>The Reporting Person is the Chief Executive Officer of Allianz Global Investors U.S. Holdings LLC, the body that performs executive oversight functions for Allianz Global Investors U.S. LLC, the Issuer's sub-adviser.  Exhibit List: Exhibit 24 - Power of Attorney.</remarks>

    <ownerSignature>
        <signatureName>/s/ Thomas J. Fuccillo, Attorney-in-Fact for Douglas Eu</signatureName>
        <signatureDate>2016-01-08</signatureDate>
    </ownerSignature>
</ownershipDocument>
