Before the advent of digital computers, sophisticated orreries were used to predict the positions and motions of astronomical bodies. Today, we are witnessing the renaissance of devices that simulate, rather than calculate, the evolution of complex many-body systems. Quantum simulators — which use one controllable quantum system to investigate the behaviour and properties of another, less accessible one — hold the promise of tackling problems that are too demanding for classical computers. Over the past few years, significant progress has been made in a number of experimental fields, as reviewed in this Insight, which also considers where quantum simulation might take us.