In active matter systems, individual constituents convert energy into non-conservative forces or motion at the microscale, leading to morphological features and transport properties that do not occur in equilibrium and that are robust against certain perturbations. In recent years, a fruitful method for analysing these features has been to use tools from topology. In this Review, we focus on topological defects and topologically protected edge modes, with an emphasis on the distinctive properties they acquire in active media. These paradigmatic examples represent two physically distinct classes of phenomena that are robust thanks to a common mathematical origin: the presence of topological invariants. Beyond active matter, our Review underscores the role of topological excitations in non-equilibrium settings of relevance, from open quantum systems to living matter.