  It is known that every proper minor-closed class of graphs has bounded
stack-number (a.k.a. book thickness and page number). While this includes
notable graph families such as planar graphs and graphs of bounded genus, many
other graph families are not closed under taking minors. For fixed $g$ and $k$,
we show that every $n$-vertex graph that can be embedded on a surface of genus
$g$ with at most $k$ crossings per edge has stack-number $\mathcal{O}(\log n)$;
this includes $k$-planar graphs. The previously best known bound for the
stack-number of these families was $\mathcal{O}(\sqrt{n})$, except in the case
of $1$-planar graphs. Analogous results are proved for map graphs that can be
embedded on a surface of fixed genus. None of these families is closed under
taking minors. The main ingredient in the proof of these results is a
construction proving that $n$-vertex graphs that admit constant layered
separators have $\mathcal{O}(\log n)$ stack-number.
