Discrete subgroups of Lie groups play a fundamental role in several areas of mathematics. In the case of SL(2,R), they are well understood and classified by the geometry of the corresponding hyperbolic surfaces. In the case of SL(n,R) with n>2, they remain more mysterious, beyond the important class of lattices (i.e. discrete subgroups of finite covolume for the Haar measure). These past twenty years, several interesting classes of discrete subgroups have emerged, which are "thinner" than lattices, more flexible, and with remarkable geometric and dynamical properties. We will present some recent developments in the subject.