Realizing topological superconductivity and Majorana zero modes in the laboratory is a major goal in condensed-matter physics. In this Review, we survey the current status of this rapidly developing field, focusing on proposals for the realization of topological superconductivity in semiconductor–superconductor heterostructures. We examine materials science progress in growing InAs and InSb semiconductor nanowires and characterizing these systems. We then discuss the observation of robust signatures of Majorana zero modes in recent experiments, paying particular attention to zero-bias tunnelling conduction measurements and Coulomb blockade experiments. We also outline several next-generation experiments probing exotic properties of Majorana zero modes, including fusion rules and non-Abelian exchange statistics. Finally, we discuss prospects for implementing Majorana-based topological quantum computation.