Given a fixed graph $H$ that embeds in a surface $Σ$, what is the maximum number of copies of $H$ in an $n$-vertex graph $G$ that embeds in $Σ$? We show that the answer is $Θ(n^{f(H)})$, where $f(H)$ is a graph invariant called the `flap-number' of $H$, which is independent of $Σ$. This simultaneously answers two open problems posed by Eppstein (1993). When $H$ is a complete graph we give more precise answers.