  We prove that for closed surfaces $M$ with Riemannian metrics without
conjugate points and genus $\geq 2$ the geodesic flow on the unit tangent
bundle $T^1M$ has a unique measure of maximal entropy. Furthermore, this
measure is fully supported on $T^1M$ and the flow is mixing with respect to
this measure. We formulate conditions under which this result extends to higher
dimensions.
