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

    <schemaVersion>X0206</schemaVersion>

    <documentType>3</documentType>

    <periodOfReport>2022-01-01</periodOfReport>

    <noSecuritiesOwned>0</noSecuritiesOwned>

    <issuer>
        <issuerCik>0001441236</issuerCik>
        <issuerName>Clearwater Paper Corp</issuerName>
        <issuerTradingSymbol>CLW</issuerTradingSymbol>
    </issuer>

    <reportingOwner>
        <reportingOwnerId>
            <rptOwnerCik>0001899915</rptOwnerCik>
            <rptOwnerName>Urlick Michael J</rptOwnerName>
        </reportingOwnerId>
        <reportingOwnerAddress>
            <rptOwnerStreet1>601 W. RIVERSIDE AVE.</rptOwnerStreet1>
            <rptOwnerStreet2>SUITE 1100</rptOwnerStreet2>
            <rptOwnerCity>SPOKANE</rptOwnerCity>
            <rptOwnerState>WA</rptOwnerState>
            <rptOwnerZipCode>99201</rptOwnerZipCode>
            <rptOwnerStateDescription></rptOwnerStateDescription>
        </reportingOwnerAddress>
        <reportingOwnerRelationship>
            <isDirector>0</isDirector>
            <isOfficer>1</isOfficer>
            <isTenPercentOwner>0</isTenPercentOwner>
            <isOther>0</isOther>
            <officerTitle>Sr. VP, GM, CPD</officerTitle>
        </reportingOwnerRelationship>
    </reportingOwner>

    <nonDerivativeTable>
        <nonDerivativeHolding>
            <securityTitle>
                <value>Common Stock</value>
            </securityTitle>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>2602</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </nonDerivativeHolding>
        <nonDerivativeHolding>
            <securityTitle>
                <value>Common Stock</value>
                <footnoteId id="F1"/>
            </securityTitle>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>569</value>
                    <footnoteId id="F2"/>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </nonDerivativeHolding>
        <nonDerivativeHolding>
            <securityTitle>
                <value>Common Stock</value>
                <footnoteId id="F1"/>
            </securityTitle>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>1589</value>
                    <footnoteId id="F3"/>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </nonDerivativeHolding>
        <nonDerivativeHolding>
            <securityTitle>
                <value>Common Stock</value>
                <footnoteId id="F1"/>
            </securityTitle>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>2193</value>
                    <footnoteId id="F4"/>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </nonDerivativeHolding>
        <nonDerivativeHolding>
            <securityTitle>
                <value>Common Stock</value>
                <footnoteId id="F1"/>
            </securityTitle>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>843</value>
                    <footnoteId id="F5"/>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </nonDerivativeHolding>
    </nonDerivativeTable>

    <derivativeTable>
        <derivativeHolding>
            <securityTitle>
                <value>Employee Right to Buy (Stock Option)</value>
            </securityTitle>
            <conversionOrExercisePrice>
                <value>61.75</value>
            </conversionOrExercisePrice>
            <exerciseDate>
                <value>2017-12-31</value>
            </exerciseDate>
            <expirationDate>
                <value>2025-02-26</value>
            </expirationDate>
            <underlyingSecurity>
                <underlyingSecurityTitle>
                    <value>Common Stock</value>
                </underlyingSecurityTitle>
                <underlyingSecurityShares>
                    <value>660</value>
                </underlyingSecurityShares>
            </underlyingSecurity>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </derivativeHolding>
        <derivativeHolding>
            <securityTitle>
                <value>Employee Right to Buy (Stock Option)</value>
            </securityTitle>
            <conversionOrExercisePrice>
                <value>38.75</value>
            </conversionOrExercisePrice>
            <exerciseDate>
                <value>2018-12-31</value>
            </exerciseDate>
            <expirationDate>
                <value>2026-02-25</value>
            </expirationDate>
            <underlyingSecurity>
                <underlyingSecurityTitle>
                    <value>Common Stock</value>
                </underlyingSecurityTitle>
                <underlyingSecurityShares>
                    <value>1311</value>
                </underlyingSecurityShares>
            </underlyingSecurity>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </derivativeHolding>
        <derivativeHolding>
            <securityTitle>
                <value>Employee Right to Buy (Stock Option)</value>
            </securityTitle>
            <conversionOrExercisePrice>
                <value>56.75</value>
            </conversionOrExercisePrice>
            <exerciseDate>
                <value>2020-03-01</value>
            </exerciseDate>
            <expirationDate>
                <value>2027-02-27</value>
            </expirationDate>
            <underlyingSecurity>
                <underlyingSecurityTitle>
                    <value>Common Stock</value>
                </underlyingSecurityTitle>
                <underlyingSecurityShares>
                    <value>846</value>
                </underlyingSecurityShares>
            </underlyingSecurity>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </derivativeHolding>
        <derivativeHolding>
            <securityTitle>
                <value>Employee Right to Buy (Stock Option)</value>
            </securityTitle>
            <conversionOrExercisePrice>
                <value>37.45</value>
            </conversionOrExercisePrice>
            <exerciseDate>
                <value>2021-03-05</value>
            </exerciseDate>
            <expirationDate>
                <value>2028-03-05</value>
            </expirationDate>
            <underlyingSecurity>
                <underlyingSecurityTitle>
                    <value>Common Stock</value>
                </underlyingSecurityTitle>
                <underlyingSecurityShares>
                    <value>1356</value>
                </underlyingSecurityShares>
            </underlyingSecurity>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </derivativeHolding>
    </derivativeTable>

    <footnotes>
        <footnote id="F1">Represents award of restricted stock units (&quot;RSUs&quot;).  RSUs may be settled only for shares of common stock on a one-for-one basis.</footnote>
        <footnote id="F2">Represents award of RSUs that will vest March 7, 2022, assuming continued employment.  During the vesting period, an amount equal to the dividends that would have been paid on the RSUs had they been in the form of common stock will be converted into additional RSUs.</footnote>
        <footnote id="F3">Represents award of RSUs that will vest on March 3, 2022, 2023 respectively, assuming continued employment.  During the vesting period, an amount equal to the dividends that would have been paid on the RSUs had they been in the form of common stock will be converted into additional RSUs.</footnote>
        <footnote id="F4">Represents award of RSUs that will vest 33%, 33%, 34% on March 4, 2022, 2023 and 2024, respectively, assuming continued employment.  During the vesting period, an amount equal to the dividends that would have been paid on the RSUs had they been in the form of common stock will be converted into additional RSUs.</footnote>
        <footnote id="F5">Represents award of RSUs that will vest 33%, 33%, 34% on July 15, 2022, 2023 and 2024, respectively, assuming continued employment.  During the vesting period, an amount equal to the dividends that would have been paid on the RSUs had they been in the form of common stock will be converted into additional RSUs.</footnote>
    </footnotes>

    <remarks></remarks>

    <ownerSignature>
        <signatureName>/s/ Michael S. Gadd, Attorney-in-Fact</signatureName>
        <signatureDate>2022-01-07</signatureDate>
    </ownerSignature>
</ownershipDocument>
