In 1756, Leidenfrost observed that water drops skittered on a sufficiently hot skillet, owing to levitation by an evaporative vapour film. Such films are stable only when the hot surface is above a critical temperature, and are a central phenomenon in boiling. In this so-called Leidenfrost regime, the low thermal conductivity of the vapour layer inhibits heat transfer between the hot surface and the liquid. When the temperature of the cooling surface drops below the critical temperature, the vapour film collapses and the system enters a nucleate-boiling regime, which can result in vapour explosions that are particularly detrimental in certain contexts, such as in nuclear power plants. The presence of these vapour films can also reduce liquid-solid drag. Here we show how vapour film collapse can be completely suppressed at textured superhydrophobic surfaces. At a smooth hydrophobic surface, the vapour film still collapses on cooling, albeit at a reduced critical temperature, and the system switches explosively to nucleate boiling. In contrast, at textured, superhydrophobic surfaces, the vapour layer gradually relaxes until the surface is completely cooled, without exhibiting a nucleate-boiling phase. This result demonstrates that topological texture on superhydrophobic materials is critical in stabilizing the vapour layer and thus in controlling--by heat transfer--the liquid-gas phase transition at hot surfaces. This concept can potentially be applied to control other phase transitions, such as ice or frost formation, and to the design of low-drag surfaces at which the vapour phase is stabilized in the grooves of textures without heating.