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

    <schemaVersion>X0306</schemaVersion>

    <documentType>4</documentType>

    <periodOfReport>2021-01-29</periodOfReport>

    <notSubjectToSection16>0</notSubjectToSection16>

    <issuer>
        <issuerCik>0000078890</issuerCik>
        <issuerName>BRINKS CO</issuerName>
        <issuerTradingSymbol>BCO</issuerTradingSymbol>
    </issuer>

    <reportingOwner>
        <reportingOwnerId>
            <rptOwnerCik>0001781833</rptOwnerCik>
            <rptOwnerName>Bossart Dominik</rptOwnerName>
        </reportingOwnerId>
        <reportingOwnerAddress>
            <rptOwnerStreet1>1801 BAYBERRY COURT</rptOwnerStreet1>
            <rptOwnerStreet2>PO BOX 18100</rptOwnerStreet2>
            <rptOwnerCity>RICHMOND</rptOwnerCity>
            <rptOwnerState>VA</rptOwnerState>
            <rptOwnerZipCode>23226</rptOwnerZipCode>
            <rptOwnerStateDescription></rptOwnerStateDescription>
        </reportingOwnerAddress>
        <reportingOwnerRelationship>
            <isDirector>0</isDirector>
            <isOfficer>1</isOfficer>
            <isTenPercentOwner>0</isTenPercentOwner>
            <isOther>0</isOther>
            <officerTitle>SVP</officerTitle>
        </reportingOwnerRelationship>
    </reportingOwner>

    <derivativeTable>
        <derivativeTransaction>
            <securityTitle>
                <value>Units</value>
            </securityTitle>
            <conversionOrExercisePrice>
                <footnoteId id="F1"/>
            </conversionOrExercisePrice>
            <transactionDate>
                <value>2021-01-29</value>
            </transactionDate>
            <deemedExecutionDate></deemedExecutionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>A</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
            </transactionCoding>
            <transactionTimeliness>
                <value></value>
            </transactionTimeliness>
            <transactionAmounts>
                <transactionShares>
                    <value>50.76</value>
                    <footnoteId id="F2"/>
                </transactionShares>
                <transactionPricePerShare>
                    <value>68.13</value>
                    <footnoteId id="F3"/>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>A</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <exerciseDate>
                <footnoteId id="F1"/>
            </exerciseDate>
            <expirationDate>
                <footnoteId id="F1"/>
            </expirationDate>
            <underlyingSecurity>
                <underlyingSecurityTitle>
                    <value>Common Stock</value>
                </underlyingSecurityTitle>
                <underlyingSecurityShares>
                    <value>50.76</value>
                    <footnoteId id="F2"/>
                </underlyingSecurityShares>
            </underlyingSecurity>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>15018.68</value>
                    <footnoteId id="F4"/>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </derivativeTransaction>
    </derivativeTable>

    <footnotes>
        <footnote id="F1">Units (each of which is the economic equivalent of one share of The Brink's Company (&quot;BCO&quot;) common stock) credited to the Reporting Person's stock incentive account under the Key Employees' Deferred Compensation Program (the &quot;Program&quot;) will settle in BCO common stock on a one-for-one basis and shall be distributed in accordance with the Reporting Person's deferral election either (1) following the Reporting Person's termination of employment with BCO or (2) on a future date selected by the Reporting Person at the time of his or her deferral election.</footnote>
        <footnote id="F2">In accordance with the terms of the Program, on the last business day of each month, compensation deferred by the Reporting Person during that month and/or any matching amounts are converted into Units and credited to the Reporting Person's stock incentive account.</footnote>
        <footnote id="F3">The number of Units credited to the Reporting Person's stock incentive account on the Transaction Date is based upon a share price of $68.13, which is the closing price of BCO common stock on the final trading day of the month in which the deferred compensation would have been payable, calculated in accordance with the terms of the Program.</footnote>
        <footnote id="F4">Reflects correction of the amount of units owned by the Reporting Person due to a clerical error in a previous form 4 filing.</footnote>
    </footnotes>

    <remarks></remarks>

    <ownerSignature>
        <signatureName>/s/ Lindsay K. Blackwood
Lindsay K. Blackwood, Attorney-in-Fact</signatureName>
        <signatureDate>2021-02-02</signatureDate>
    </ownerSignature>
</ownershipDocument>
