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 $nge |V(H)|$. We discover that the chromatic number of $H$ does not play a role, as in the celebrated ErdH{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 $tge2$ and $rge 3$ are integers. However, determining $ex_{_mathcal{P}}(n,H)$, when $H$ is a planar subcubic graph, remains wide open.
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