Abstract: We consider the inverse boundary value problem that arises when a spatially variable sound speed inside a body is probed using acoustic waves that are produced and recorded on the surface of the body. This serves as a model problem e.g. for seismic imaging. In this talk, we consider the wave equation for the Laplace operator of a smooth compact Riemannian manifold with boundary and show that acoustic measurements with sources and receivers on disjoints sets on the boundary determine the manifold uniquely assuming that the wave equation is exactly controllable from the set of sources. The exact controllability can be characterized in terms of the billiard flow of the manifold. We will also consider the problem to determine lower order terms in a wave equation.