These lectures relates to applications of methods of the modern theory of
integrable systems to geodesic flows on Riemannian manifolds. We expose 
some basics of methods developed for integrating geodesic flows, topological 
obstructions to integrability in terms of tame (in particular, real analytic) 
integrals of motion and some recent results on flows which are completely 
integrable only by means of wild smooth integrals of motion.