Confounder selection is perhaps the most important step in the design of observational studies. A number of criteria, often with different objectives and approaches, have been proposed, and their validity and practical value have been debated in the literature. Here, we provide a unified review of these criteria and the assumptions behind them. We list several objectives that confounder selection methods aim to achieve and discuss the amount of structural knowledge required by different approaches. Finally, we discuss limitations of the existing approaches and implications for practitioners.