This paper gives a proof of the existence and multiplicity of periodic solutions to Hamiltonian systems of the form \[ ( A ) { m i q ¨ i + ∂ V ∂ q i ( t , q ) = 0 q ( t + T ) = q ( t ) , ∀ t ∈ ℜ . ({\text {A}})\quad {\text { }}\left \{ {\begin {array}{*{20}{c}} {{m_i}{{\ddot q}_i} + \frac {{\partial V}} {{\partial {q_i}}}(t,q) = 0} \\ {q(t + T) = q(t),\quad \forall t \in \Re .} \\ \end {array} } \right . \] where q i ∈ ℜ ℓ , ℓ ⩾ 3 , 1 ⩽ i ⩽ n , q = ( q 1 , … , q n ) {q_i} \in {\Re ^\ell },\ell \geqslant 3,1 \leqslant i \leqslant n,q = ({q_1}, \ldots ,{q_n}) and with V i j ( t , ξ ) {V_{ij}}(t,\xi ) T T -periodic in t t and singular in ξ \xi at ξ = 0 \xi = 0 Under additional hypotheses on V V , when (A) is posed as a variational problem, the corresponding functional, I I , is shown to have an unbounded sequence of critical values if the singularity of V V at 0 0 is strong enough. The critical points of I I are classical T T -periodic solutions of (A). Then, assuming that I I has only non-degenerate critical points, up to translations, Morse type inequalities are proved and used to show that the number of critical points with a fixed Morse index k k grows exponentially with k k , at least when k ≡ 0 , 1 ( mod ℓ − 2 ) k \equiv 0,1( \mod \ell - 2) . The proof is based on the study of the critical points at infinity done by the author in a previous paper and generalizes the topological arguments used by A. Bahri and P. Rabinowitz in their study of the 3 3 -body problem. It uses a recent result of E. Fadell and S. Husseini on the homology of free loop spaces on configuration spaces. The detailed proof is given for the 4 4 -body problem then generalized to the n n -body problem.