Dynamical zeta functions of Selberg, Smale and Ruelle are analogous to the Riemann zeta function with the product over primes replaced by products over closed orbits of Anosov flows. In 1967 Smale conjectured that these zeta functions should be meromorphic but admitted "that a positive answer would be a little shocking". Nevertheless the continuation was proved in 2012 by Giulietti-Liverani-Pollicott. In my lecture I will present a proof of this result obtained by Dyatlov and myself and inspired by a trace formula of Guillemin and by recent work of Faure-Sjoestrand. It is based on a simple idea involving wave front sets and propagation of singularities: we apply methods of microlocal analysis to the generator of the flow, an in particular propagation results due to Duistermaat-Hoermander, Melrose and Vasy.