  An index theory for projective families of elliptic pseudodifferential
operators is developed. The topological and the analytic index of such a family
both take values in twisted K-theory of the parametrizing space, X. The main
result is the equality of these two notions of index when the twisting class is
in the torsion subgroup of H^3(X;Z). The Chern character of the index class is
then computed.
