  Our aim is to determine the lower central series (LCS) and derived series
(DS) for the braid groups of the sphere and of the finitely-punctured sphere.
We show that for all n (resp. all n\geq 5), the LCS (resp. DS) of the n-string
braid group B\_n(S^2) is constant from the commutator subgroup onwards, and
that \Gamma\_2(B\_4(S^2)) is a semi-direct product of the quaternion group by a
free group of rank 2. For n=4, we determine the DS of B\_4(S^2), as well as its
quotients. For n \geq 1, the class of m-string braid groups B\_m(S^2) \
{x\_1,...,x\_n} of the n-punctured sphere includes the Artin braid groups B\_m,
those of the annulus, and certain Artin and affine Artin groups. We extend
results of Gorin and Lin, and show that the LCS (resp. DS) of B\_m is
determined for all m (resp. for all m\neq 4). For m=4, we obtain some elements
of the DS. When n\geq 2, we prove that the LCS (resp. DS) of B\_m(S^2) \
{x\_1,...,x\_n} is constant from the commutator subgroup onwards for all m\geq
3 (resp. m\geq 5). We then show that B\_2(S^2\{x\_1,x\_2}) is residually
nilpotent, that its LCS coincides with that of Z\_2*Z, and that the
\Gamma\_i/\Gamma\_{i+1} are 2-elementary finitely-generated groups. For m\geq 3
and n=2, we obtain a presentation of the derived subgroup and its
Abelianisation. For n=3, we see that the quotients \Gamma\_i/\Gamma\_{i+1} are
2-elementary finitely-generated groups.
