  Given a graph $H$, a graph is $H$-free if it does not contain $H$ as a
subgraph. We continue to study the topic of "extremal" planar graphs, that is,
how many edges can an $H$-free planar graph on $n$ vertices have? We define
$ex_{_\mathcal{P}}(n,H)$ to be the maximum number of edges in an $H$-free
planar graph on $n $ vertices. We first obtain several sufficient conditions on
$H$ which yield $ex_{_\mathcal{P}}(n,H)=3n-6$ for all $n\ge |V(H)|$. We
discover that the chromatic number of $H$ does not play a role, as in the
celebrated Erd\H{o}s-Stone Theorem. We then completely determine
$ex_{_\mathcal{P}}(n,H)$ when $H$ is a wheel or a star. Finally, we examine the
case when $H$ is a $(t, r)$-fan, that is, $H$ is isomorphic to $K_1+tK_{r-1}$,
where $t\ge2$ and $r\ge 3$ are integers. However, determining
$ex_{_\mathcal{P}}(n,H)$, when $H$ is a planar subcubic graph, remains wide
open.
