Abstract
To every semigroup presentation P=⟨Σ∣R⟩ and every baseword w∈Σ+ can be associated a diagram group D(P,w), defined as the fundamental group of the Squier complex S(P,w). Roughly speaking, D(P,w) encodes the lack of asphericity of P. Examples of diagram groups include Thompson's group F, the wreath product Z≀Z, the pure planar braid groups, and various right-angled Artin groups. This survey aims at summarising what is known about the family of diagram groups.