Many laser systems show self-pulsations with a large amplitude that are born suddenly in a homoclinic bifurcation. Just before the onset of these self-pulsations the laser is excitable where the excitability threshold is formed by the stable manifold of a saddle point. We show that there exists a special configuration, a codimension-two bifurcation called a non-central saddle-node homoclinic orbit, that acts as an organising centre of excitability in lasers. It is the key to understanding excitability in laser systems as diverse as lasers with saturable absorbers, lasers with optical injection and lasers with optical feedback. © 2002 Elsevier Science B.V. All rights reserved.