  We classify gradings by arbitrary abelian groups on the classical simple Lie
superalgebras $P(n)$, $n \geq 2$, and on the simple associative superalgebras
$M(m,n)$, $m, n \geq 1$, over an algebraically closed field: fine gradings up
to equivalence and $G$-gradings, for a fixed group $G$, up to isomorphism. As a
corollary, we also classify up to isomorphism the $G$-gradings on the classical
Lie superalgebra $A(m,n)$ that are induced from $G$-gradings on $M(m+1,n+1)$.
In the case of Lie superalgebras, the characteristic is assumed to be $0$.
