The nonlinear stability of spacetimes satisfying Einstein equations is understood only under very special circumstances: small perturbations of ``known'' solutions or solutions enjoying certain symmetry restrictions. Gravitational attraction may lead to collapse or dispersion, and it is extremely challenging to cope with these phenomena mathematically. In this lecture, I will present a selection of recent advances on the global evolution of self-gravitating fields and on the theory of f(R) gravity. Blog: philippelefloch.org