We characterize the deformation mechanisms of crystalline and non-crystalline dry liquid foams. Crystalline foams deform by dislocation glide. We identify the operating dislocations in 2D and 3D foams and the mechanism by which they glide. In 2D foams the dislocations are pairs of bubbles with 5 and 7 sides. In 3D foams they are (line) defects consisting of a row of 12- and 16-faced bubbles. Dislocations nucleate when the strain reaches a critical value for which the arrangement of the bubbles is no longer stable. Dislocation glide then occurs by an avalanche of elementary topological changes. Disordered foams also deform by avalanches of topological changes, similar to the ones that occur in dislocation glide in ordered foams. We show, for the first time, experimental force-displacement curves obtained in uniaxial tension-compression tests of a 3 D foam. The curves show jumps in the applied force associated with the topological changes.