This book chapter is published open access.
Spectral theory of one-dimensional discrete one-frequency Schrödinger operators is a field with the origins in and strong ongoing ties to physics. It features a fascinating competition between randomness (ergodicity) and order (periodicity), which is often resolved on a deep arithmetic level. This leads to an especially rich spectrum of phenomena, many of which we are only beginning to understand. The corresponding analysis involves, in particular, dealing with small denominator problems. It has led to the development of non-KAM methods in this traditionally KAM domain, and to results completely unattainable by the old techniques, also in a number of other settings. This article accompanies the author’s lecture at the International Congress of Mathematicians 2022. It covers several related recent developments.