Trace formulas have been one of the most powerfull tools in spectral geometry. This article is an invitation to spectral geometry via trace formulas paradigm. The main goal of the present survey is to give some classical results concerning trace formulas in Riemannian geometry and to present some nice applications in conformal geometry and in dynamical systems. More precisely, we explain very recent results concerning the study of the geodesic flows on manifolds with negative variable curvature via the semi-classical zeta functions for contact Anosov flows.