In this paper we give topological lower bounds on the number of
periodic and of closed trajectories in strictly convex smooth
billiards $T\subset \mathbf {R}\sp {m+1}$. Namely, for given $n$, we
estimate the number of $n$-periodic billiard trajectories in $T$ and
also estimate the number of billiard trajectories which start and end
at a given point $A\in \partial T$ and make a prescribed number n of
reflections at the boundary $\partial T$ of the billiard domain. We use
variational reduction, admitting a finite group of symmetries, and
apply a topological approach based on equivariant Morse and
Lusternik-Schnirelman theories.