  Adaptive mesh refinement techniques are nowadays an established and powerful
tool for the numerical discretization of PDE's. In recent years, wavelet bases
have been proposed as an alternative to these techniques. The main motivation
for the use of such bases in this context is their good performances in data
compression and the approximation theoretic foundations which allow to analyze
and optimize these performances. We shall discuss these theoretical
foundations, as well as one of the approaches which has been followed in
developing efficient adaptive wavelet solvers. We shall also discuss the
similarities and differences between wavelet methods and adaptive mesh
refinement.
