Few syntactic phenomena are relevant to as many areas of linguistic theory as auxiliary selection – the alternation between auxiliaries in periphrastic constructions. Standing at the intersection between syntax, lexical and clausal semantics and morphology, it has been the subject of intense research since the late 1970s and has played a role in several important theoretical developments and debates. Auxiliary selection also poses a number of empirical questions. Even within a single language, the correct description of the alternation can be controversial, and the extensive cross‐linguistic variation has turned out to be rather difficult to characterize in a systematic way. In this article, I give a survey of how our understanding of auxiliary selection has developed, both on the empirical and the theoretical side. I also include a brief discussion of work on how patterns of selection arise and change over time.