Generalized Turán results for matchings

Given graphs $H$ and $F$, the generalized Turán number $\mathrm{e}\mathrm{x}(n,H,F)$ is the largest number of copies of $H$ in $n$-vertex $F$-free graphs. We study the case when either $H$ or $F$ is a matching. We obtain several asymptotic and exact results.