  This survey is based on a series of talks I gave at the conference "Dynamical
systems and diophantine approximation" at l'Instut Henri Poincar\'e in June
2003. I will present asymptotic results (transitivity, ergodicity, weak-mixing)
for billiards based on the approximation technique developed by Katok and
Zemlyakov. I will also present approximation techniques which allow to prove
the abundance of periodic trajectories in certain irrational polygons.
