  For graphs $H$ and $F$, the generalized Tur\'an number $ex(n,H,F)$ is the
largest number of copies of $H$ in an $F$-free graph on $n$ vertices. We
consider this problem when both $H$ and $F$ have at most four vertices. We give
sharp results in almost all cases, and connect the remaining cases to
well-known unsolved problems. Our main new contribution is applying the
progressive induction method of Simonovits for generalized Tur\'an problems.
