Group gradings on the Lie and Jordan superalgebras $Q(n)$

We classify gradings by arbitrary abelian groups on the classical simple Lie and Jordan superalgebras $Q(n)$, $n\ge 2$, over an algebraically closed field of characteristic different from $2$ (and not dividing $n+1$ in the Lie case): fine gradings up to equivalence and $G$-gradings, for a fixed group $G$, up to isomorphism.