  We extend the Marcus-Schaeffer bijection between orientable rooted bipartite
quadrangulations (equivalently: rooted maps) and orientable labeled one-face
maps to the case of all surfaces, that is orientable and non-orientable as
well. This general construction requires new ideas and is more delicate than
the special orientable case, but it carries the same information. In
particular, it leads to a uniform combinatorial interpretation of the counting
exponent $\frac{5(h-1)}{2}$ for both orientable and non-orientable rooted
connected maps of Euler characteristic $2-2h$, and of the algebraicity of their
generating functions, similar to the one previously obtained in the orientable
case via the Marcus-Schaeffer bijection. It also shows that the renormalization
factor $n^{1/4}$ for distances between vertices is universal for maps on all
surfaces: the renormalized profile and radius in a uniform random pointed
bipartite quadrangulation on any fixed surface converge in distribution when
the size $n$ tends to infinity. Finally, we extend the Miermont and
Ambj{\o}rn-Budd bijections to the general setting of all surfaces. Our
construction opens the way to the study of Brownian surfaces for any compact
2-dimensional manifold.
