Large scale geometry systematically ignores properties of a metric space that are present only at a fixed finite scale. For example, a bounded metric space appears to be a point when viewed from sufficiently far away, and the real numbers and integers become indistinguishable. Ideas and methods from large scale geometry play an important role in the study of several important problems in high dimensional topology and operator algebra theory, including the Borel, Baum-Connes and Novikov conjectures. In joint work in progress with Romain Tessera and Guoliang Yu we introduce the notion of finite decomposition complexity, a large scale property of metric spaces that generalizes the notion of finite asymptotic dimension. In the talk we shall focus on giving examples of groups with finite decomposition complexity. We may, as time allows, discuss applications to questions of topological rigidity.