The celebrated 1983 paper by Thouless and coworkers (TKNN) marks the first appearance of topology in nonrelativistic quantum mechanics and in condensed matter physics. In 1988 Haldane proposed a model material (named by us “Haldanium”) which has two phases, one trivial and one very exotic: what discriminates between the two is the topology of the electronic ground state. Haldane’s work opened a new avenue, and several topological materials were later synthetised; the current literature is flooded with “topological” papers: experimental, theoretical, and computational.

In this talk I will convey the main message of the two papers quoted above by keeping the technical level as low as possible; in the final part I will also give a flavour of some recent developments.