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We consider the (pure) braid groups B_{n}(M) and P_{n}(M), where M is the2-sphere S^2 or the real projective plane RP^2. We determine the minimalcardinality of (normal) generating sets X of these groups, first when there isno restriction on X, and secondly when X consists of elements of finite order.This improves on results of Berrick and Matthey in the case of S^2, and extendsthem in the case of RP^2. We begin by recalling the situation for the Artinbraid groups. As applications of our results, we answer the correspondingquestions for the associated mapping class groups, and we show that for M=S^2or RP^2, the induced action of B_n(M) on H_3 of the universal covering of the nth configuration space of M is trivial.

Minimal generating and normally generating sets for the braid and mapping class groups of the disc, the sphere and the projective plane