Species replace each other in space and time because a single species cannot adapt to all environments. Failure to adapt to a changing environment in time leads to extinction, while failure to adapt to a changing environment in space generates the finite ranges that are a feature of all species. Understanding patterns of biodiversity therefore depends on understanding what limits adaptation to ecological change both at range margins, and in all populations over time. Here, we review theoretical models and empirical data concerning what determines maximum rates of adaptation in time or in space, and try to draw parallels between them. The key issues for both types of model are the amount of additive genetic variation available in the direction demanded by selection, which may be low even if variation in single traits is high, and the demographic costs incurred by populations while they adapt to changing optima. Not only is measurement of these parameters data and labour intensive, they are also likely to vary from population to population and generation to generation, even for a single species. For this reason, it remains difficult to predict how easily, or how rapidly, evolution will occur in response to ecological change in a given situation