My talk will cover a broad range of topics highlighting interactions between abstract Turing computability and classification problems in topology. The subject is wide-ranging and can be roughly divided into two interconnected directions: (1) computable (“constructive”) aspects of topology, and (2) applications of formal models of computation to problems in topology that may initially appear unrelated to computation. As will be seen, these two directions are closely linked at a technical level, and no clear dividing line can be drawn between them.