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Humans are highly efficient learners, with the ability to grasp the meaning of a new concept from just a few examples. Unlike popular computer vision systems, humans can flexibly leverage the compositional structure of the visual world, understanding new concepts as combinations of existing concepts. In the current paper, we study how people learn different types of visual compositions, using abstract visual forms with rich relational structure. We find that people can make meaningful compositional generalizations from just a few examples in a variety of scenarios, and we develop a Bayesian program induction model that provides a close fit to the behavioral data. Unlike past work examining special cases of compositionality, our work shows how a single computational approach can account for many distinct types of compositional generalization.