  For two given graphs $G$ and $H$ the planar Ramsey number $PR(G,H)$ is the
smallest integer $n$ such that every planar graph $F$ on $n$ vertices either
contains a copy of $G$, or its complement contains a copy of $H$. In this
paper, we first characterize some structural properties of $C_4$-free planar
graphs, and then we determine all planar Ramsey numbers $PR(C_4, W_n)$, for
$n\ge 3$.
