The inevitability of sphalerons in field theory

The topological structure of field theory often makes inevitable the existence of stable and unstable localized solutions of the field equations. These are minima and saddle points of the energy. Saddle point solutions occurring this way are known as sphalerons, and the most interesting one is in the electroweak theory of coupled W, Z and Higgs bosons. The topological ideas underpinning sphalerons are reviewed here. This article is part of a discussion meeting issue 'Topological avatars of new physics'.