Categorization is a fundamental function of minds, with wide ranging implications for the rest of the cognitive system. In humans, categories are shared and communicated between minds, thus requiring explanations at the population level. In this paper, we discuss the current state of research on the cultural evolution of categorization. We begin by delineating key properties of categories in need of evolutionary explanation. We then review computational modeling and laboratory studies of category evolution, including their major insights and limitations. Finally, we discuss remaining challenges for understanding the cultural evolution of categorization.