Trade-offs are a core component of many evolutionary models, particularly those dealing with the evolution of life histories. In the present paper, we identify four topics of key importance for studies of the evolutionary biology of trade-offs. First, we consider the underlying concept of 'constraint'. We conclude that this term is typically used too vaguely and suggest that 'constraint' in the sense of a bias should be clearly distinguished from 'constraint' in the sense of proscribed combinations of traits or evolutionary trajectories. Secondly, we address the utility of the acquisition-allocation model (the 'Y-model'). We find that, whereas this model and its derivatives have provided new insights, a misunderstanding of the pivotal equation has led to incorrect predictions and faulty tests. Thirdly, we ask how trade-offs are expected to evolve under directional selection. A quantitative genetic model predicts that, under weak or short-term selection, the intercept will change but the slope will remain constant. Two empirical tests support this prediction but these are based on comparisons of geographic populations: more direct tests will come from artificial selection experiments. Finally, we discuss what maintains variation in trade-offs noting that at present little attention has been given to this question. We distinguish between phenotypic and genetic variation and suggest that the latter is most in need of explanation. We suggest that four factors deserving investigation are mutation-selection balance, antagonistic pleiotropy, correlational selection and spatio-temporal variation, but as in the other areas of research on trade-offs, empirical generalizations are impeded by lack of data. Although this lack is discouraging, we suggest that it provides a rich ground for further study and the integration of many disciplines, including the emerging field of genomics.