Asymptotic dimension is a large scale invariant of a metric space (e.g. a group) introduced by Gromov. It parallels in many ways the classical (Hurewicz-Wallman) covering dimension of topological spaces. When asymptotic dimension of a group is finite, standard (Novikov type) conjectures follow. In the talk I will explain the definition and how one proves finiteness for some well known groups (for example, hyperbolic groups). In the last part of the talk I will outline the main ideas in the recent work, joint with Ken Bromberg and Koji Fujiwara, that mapping class groups have finite asymptotic dimension.