  We prove an index theorem for families of linear periodic Hamiltonian
systems, which is reminiscent of the Atiyah-Singer index theorem for
selfadjoint elliptic operators. For the special case of one-parameter families,
we compare our theorem with a classical result of Salamon and Zehnder. Finally,
we use the index theorem to study bifurcation of branches of periodic solutions
for families of nonlinear Hamiltonian systems.
