The key aspects of the physics of unstable flows in compressors are described. Operating at part-load can cause serious instabilities in the compressor flow, even leading to damage to the compressor. Different types of unsteady flow can be categorised as surge, rotating stall and hysteresis, and these depend on both the compressor and the process to which it delivers the flow. The key parameter in the system dynamics that is used to measure the likelihood of rotating stall or surge is a stability parameter known as the Greitzer B parameter. The onset of instability can happen in two different ways, known as modes and spikes. The consequence of instability on the operating range is described, and field experience shows that the operating range reduces with higher tip-speed Mach numbers and larger work coefficients. The system requirements can be categorised in terms of the pressure versus volume characteristics of the process. Methods to extend the stable operating range of compressors by control with variable speed, variable geometry, passive recirculation systems and other regulation devices are described.