The talk deals with the wave equation on a Riemannian manifold. The propagator is the operator which maps initial conditions to a solution of the wave equation. The goal is to construct the propagator explicitly, modulo an integral operator with infinitely smooth kernel. Here by "explicitly" we mean reducing the problem to integration of ordinary differential equations. It has been known since the 1950s that this goal can be achieved using microlocal techniques, however in its standard version this construction is local in space and in time and involves taking compositions of operators. It turns out that the propagator can be written as a single oscillatory integral, global in space and in time, and that this can be done in an invariant geometric fashion. The results presented in the talk are a development of earlier results of Ari Laptev, Yuri Safarov and the speaker. This is joint work with Matteo Capoferri and Michael Levitin.