Dynamics beyond uniform hyperbolicity : a global geometric and probabilistic perspective The notion of uniform hyperbolicity introduced by Steve Smale in the early sixties unified important developments and led to a remarkably successful theory for a large class of systems: uniformly hyperbolic systems often exhibit complicated evolution which nevertheless is now rather well understood both geometrically and statistically. Another revolution has been taking place in the last couple of decades as one tries to build a global theory for most dynamical systems recovering as much as possible of the conclusions of the uniformly hyperbolic case in great generality. This book aims to put such recent developments in a unified perspective and to point out open problems and likely directions for further progress. It is aimed at researchers both young and senior willing to get a quick yet broad view of this part of dynamics. Main ideas methods and results are discussed at variable degrees of depth with references to the original works for details and complementary information