Ultracold neutrons (UCN), neutrons with speeds less than 7 ms-1, have the surprising property that they are reflected at normal incidence by many solid materials and can be stored as a neutron gas in a 'bottle' for a time comparable with the neutron's spontaneous lifetime of 937 s. Their flight paths are bent because of gravity, they are brought to rest after vertical motion of only 2 m, and a magnetic field of 5 tesla, interacting with the neutron magnetic moment, is sufficient to reverse the direction of motion. This review contains an account of the way UCN interact with surfaces in which the interatomic spacing is much less than the neutron deBroglie wavelength, the construction of UCN sources, and some of the experiments which have already been carried out to study UCN interactions with materials, gravitation, magnetic and electric fields.