We present a brief introduction to the orbital angular momentum of light, the subject of our theme issue and, in particular, to the developments in the 13 years following the founding paper by Allenet al.(Allenet al.1992Phys. Rev. A45, 8185 (doi:10.1103/PhysRevA.45.8185)). The papers by our invited authors serve to bring the field up to date and suggest where developments may take us next.This article is part of the themed issue ‘Optical orbital angular momentum’.