  This is an expository paper devoted to the Morse index of Chaperon's
generating families of Hamiltonian diffeomorphisms. After reviewing the
construction of such generating families, we present Bott's iteration theory in
this setting: we study how the Morse index of a critical point corresponding to
an iterated periodic orbit depends on the order of iteration of the orbit. We
also investigate the precise dependence of the Morse index from the choice of
the generating family associated to a given Hamiltonian diffeomorphism, which
will allow to see the Morse index as a Maslov index for the linearized
Hamiltonian flow in the symplectic group. We will conclude the survey with a
proof that the classical Morse index from Tonelli Lagrangian dynamics coincides
with the Maslov index.
