  We introduce the concept of matching powers of monomial ideals. Let $I$ be a
monomial ideal of $S=K[x_1,\dots,x_n]$, with $K$ a field. The $k$th matching
power of $I$ is the monomial ideal $I^{[k]}$ generated by the products
$u_1\cdots u_k$ where $u_1,\dots,u_k$ is a monomial regular sequence contained
in $I$. This concept naturally generalizes that of squarefree powers of
squarefree monomial ideals. We study depth and regularity functions of matching
powers of monomial ideals and edge ideals of weighted oriented graphs. We show
that the last nonvanishing power of a quadratic monomial ideal is always
polymatroidal and thus has a linear resolution. When $I$ is a non-quadratic
edge ideal of a weighted oriented forest, we characterize when $I^{[k]}$ has a
linear resolution.
