  Fix graphs $F$ and $H$ and let $ex(n,H,F)$ denote the maximum possible number
of copies of the graph $H$ in an $n$-vertex $F$-free graph. The systematic
study of this function was initiated by Alon and Shikhelman [{\it J. Comb.
Theory, B}. {\bf 121} (2016)]. In this paper, we give new general bounds
concerning this generalized Tur\'an function. We also determine
$ex(n,P_k,K_{2,t})$ (where $P_k$ is a path on $k$ vertices) and
$ex(n,C_k,K_{2,t})$ asymptotically for every $k$ and $t$. For example, it is
shown that for $t \geq 2$ and $k\geq 5$ we have
$ex(n,C_k,K_{2,t})=\left(\frac{1}{2k}+o(1)\right)(t-1)^{k/2}n^{k/2}$. We also
characterize the graphs $F$ that cause the function $ex(n,C_k,F)$ to be linear
in $n$. In the final section we discuss a connection between the function
$ex(n,H,F)$ and Berge hypergraph problems.
