  Let n\geq 3. We classify the finite groups which are realised as subgroups of
the sphere braid group B_n(S^2). Such groups must be of cohomological period 2
or 4. Depending on the value of n, we show that the following are the maximal
finite subgroups of B_n(S^2): Z_{2(n-1)}; the dicyclic groups of order 4n and
4(n-2); the binary tetrahedral group T_1; the binary octahedral group O_1; and
the binary icosahedral group I. We give geometric as well as some explicit
algebraic constructions of these groups in B_n(S^2), and determine the number
of conjugacy classes of such finite subgroups. We also reprove Murasugi's
classification of the torsion elements of B_n(S^2), and explain how the finite
subgroups of B_n(S^2) are related to this classification, as well as to the
lower central and derived series of B_n(S^2).
