I will discuss work in progress aimed towards defining contact homology using "virtual" holomorphic curve counting techniques.

In an ordinary billiard trajectories of a Hamiltonian system are reflected after a collision with a hypersurface (scatterer). If the scatterer has codimension more than one, we say that the billiard is degenerate. Degenerate billiards appear e.g. as limits of systems with singularities in celestial mechanics. We discuss the existence of trajectories of such systems shadowing trajectories of a degenerate billiard.