  This paper is a review on recently found connection between geodesically
equivalent metrics and integrable geodesic flows.
  Suppose two different metrics on one manifold have the same geodesics. We
show that then the geodesic flows of these metrics admit sufficiently many
integrals (of a special form) in involution and vice versa. The quantum version
of this result is also true. As a corollary we have that the topology of the
manifold (with two different metrics with the same geodesics) must be
sufficiently simple. Also we have that the non-proportionality of the metrics
at a point induces the non-proportionality of the metrics at almost all points.
