  Let M be a compact surface, either orientable or non-orientable. We study the
lower central and derived series of the braid and pure braid groups of M in
order to determine the values of n for which B\_n(M) and P\_n(M) are residually
nilpotent or residually soluble. First, we solve this problem for the case
where M is the 2-torus. We then give a general description of these series for
an arbitrary semi-direct product that allows us to calculate explicitly the
lower central series of P\_2(K), where K is the Klein bottle, and to give an
estimate for the derived series of P\_n(K). Finally, if M is a non-orientable
compact surface without boundary, we determine the values of n for which
B\_n(M) is residually nilpotent or residually soluble in the cases that were
not already known in the literature.
