"Projective transformation of pseudo-Riemannian manifolds: rigidity of Einstein manifolds and Lichnerowicz conjecture" Abstract Two metrics g and g' are geodesically equivalent, if every g-geodesic, after the appro- priate reparameterisation, is a g'-geodesic. In the present talk, which is in particular based on recent joint paper with Kiosak (http://xxx.lanl.gov/abs/0806.3169), I will consider geodesic equivalence of pseudo-Riemannian metrics such that the metric g is Einstein. The main result of the talk gives a complete answer to a question posed by Weyl and Petrov and is Theorem: Let g is an Einstein metric on a 4-dimensional M. If g' is geodesically equivalent to g, then it is affine equivalent to g, i.e., g and g' have the same Levi-Civita connection. The proof of this theorem is nontrivial and contains new ideas. The rest of the talk is devoted to application of these new ideas to different questions in the Riemannian and pseudo-Riemannian geometry. The big plan is to prove the conformal rigidity of Einstein metrics, nonexistence of decomposable cones over closed pseudo-Riemannian manifolds, to prove an important partial case of the projective Lichnerowicz conjecture, and to solve a classical problem explicitly stated by Sophus Lie, but I will be happy if I manage to fulfill only part of these.