The search for materials showing large caloric effects close to room temperature has become a challenge in modern materials physics and it is expected that such a class of materials will provide a way to renew present cooling devices that are based on the vapour compression of hazardous gases. Up to now, the most promising materials are giant magnetocaloric materials. The discovery of materials showing a giant magnetocaloric effect at temperatures close to ambient has opened up the possibility of using them for refrigeration(1-3). As caloric effects refer to the isothermal entropy change achieved by application of an external field, several caloric effects can take place on tuning different external parameters such as pressure and electric field. Indeed the occurrence of large electrocaloric(4,5) and elastocaloric(6) effects has recently been reported. Here we show that the application of a moderate hydrostatic pressure to a magnetic shape-memory alloy gives rise to a caloric effect with a magnitude that is comparable to the giant magnetocaloric effect reported in this class of materials. We anticipate that similar barocaloric effects will occur in many giant-magnetocaloric materials undergoing magnetostructural transitions involving a volume change.