Pollicott--Ruelle resonances are the modes appearing in correlations for Anosov flows and as singularities of dynamical zeta functions. A resonance free strip implies exponential decay of correlations and, thanks to the work of Dolgopyat, Liverani and Tsujii, it is known to exist for contact Anosov flows. Using methods of microlocal analysis and scattering theory we show that the size of that strip is finite for any Anosov flow, that is, there exist strips with infinite number of resonances. (Joint work with Long Jin and Frederic Naud).