Shape memory alloys (SMA) are a class of ferroic materials which undergo a structural (martensitic) transition where the associated ferroic property is a lattice distortion (strain). The sensitiveness of the transition to the conjugated external field (stress), together with the latent heat of the transition, gives rise to giant mechanocaloric effects. In non-magnetic SMA, the lattice distortion is mostly described by a pure shear and the martensitic transition in this family of alloys is strongly affected by uniaxial stress, whereas it is basically insensitive to hydrostatic pressure. As a result, non-magnetic alloys exhibit giant elastocaloric effects but negligible barocaloric effects. By contrast, in a number of magnetic SMA, the lattice distortion at the martensitic transition involves a volume change in addition to the shear strain. Those alloys are affected by both uniaxial stress and hydrostatic pressure and they exhibit giant elastocaloric and barocaloric effects. The paper aims at providing a critical survey of available experimental data on elastocaloric and barocaloric effects in magnetic and non-magnetic SMA. This article is part of the themed issue ‘Taking the temperature of phase transitions in cool materials’.