  In this paper, we make use of the relations between the braid and mapping
class groups of a compact, connected, non-orientable surface N without boundary
and those of its orientable double covering S to study embeddings of these
groups and their (virtual) cohomological dimensions. We first generalise
results of Birman and Chillingworth and of Gon\c{c}alves and Guaschi to show
that the mapping class group MCG(N ; k) of N relative to a k-point subset
embeds in the mapping class group MCG(S; 2k) of S relative to a 2k-point
subset. We then compute the cohomological dimension of the braid groups of all
compact, connected aspherical surfaces without boundary. Finally, if the genus
of N is greater than or equal to 2, we give upper bounds for the virtual
cohomological dimension of MCG(N ; k).
