  The sphere $S^2$ and the torus $T^2$ are the only closed connected surfaces
for which higher topological complexities are known (for each
$n\in\{2,3,...\}\subset\mathbb{N}$, $\mathrm{TC}_n(S^2)=n$ and
$\mathrm{TC}_n(T^2)=2n-2$). This text aims to find topological complexities for
most other closed connected surfaces. For all but $S^2$, $T^2$, the projective
plane ($\mathbb{R}\mathbb{P}^2$) and the Klein bottle the $n$-th topological
complexity is $2n$.
