  We determine the lower central and derived series of the n-string braid
groups B_n(RP^2) of the real projective plane. We are motivated in part by the
study of Fadell-Neuwirth short exact sequences, but the problem is interesting
in its own right. For n=1,2, B_n(RP^2) is finite and its lower central and
derived series are known. If n>2 (resp. n>4), we show that the lower central
(resp. derived) series of B_n(RP^2) is constant from the commutator subgroup
onwards, and we exhibit a presentation of the commutator subgroup. In the
exceptional cases n=3,4, we determine explicitly the complete derived series of
B_3(RP^2), we calculate the derived series of B_4(RP^2) up to and including its
fifth term, and we obtain many of the derived series quotients in these two
cases.
