Abstract: Topological insulators are materials where the bulk propagation of waves is twisted in a particular, topological way. At an interface between systems with different topologies, edge states with very special properties appear, that may for example serve as robust unidirectional wave guides. Fascinating from the point of view of mathematics, such systems are also highly relevant from the experimental point of view, and were realized (indeed) in solid-sate electronic systems, but also with light, sound, or cold atoms. I will review such phenomena, with an highlight on the bulk classification of topological "effective evolutions" which describe the propagation of waves in such peculiar media.