  In a generalized Tur\'an problem, we are given graphs $H$ and $F$ and seek to
maximize the number of copies of $H$ in an $F$-free graph of order $n$. We
consider generalized Tur\'an problems where the host graph is planar. In
particular we obtain the order of magnitude of the maximum number of copies of
a fixed tree in a planar graph containing no even cycle of length at most
$2\ell$, for all $\ell$, $\ell \geq 1$. We obtain the order of magnitude of the
maximum number of cycles of a given length in a planar $C_4$-free graph. An
exact result is given for the maximum number of $5$-cycles in a $C_4$-free
planar graph. Multiple conjectures are also introduced.
