Recently, a series of unusual earthquake phenomena have been discovered, including deep episodic tremor, low-frequency earthquakes, very-low-frequency earthquakes, slow slip events and silent earthquakes. Each of these has been demonstrated to arise from shear slip, just as do regular earthquakes, but with longer characteristic durations and radiating much less seismic energy. Here we show that these slow events follow a simple, unified scaling relationship that clearly differentiates their behaviour from that of regular earthquakes. We find that their seismic moment is proportional to the characteristic duration and their moment rate function is constant, with a spectral high-frequency decay of f(-1). This scaling and spectral behaviour demonstrates that they can be thought of as different manifestations of the same phenomena and that they comprise a new earthquake category. The observed scale dependence of rupture velocity for these events can be explained by either a constant low-stress drop model or a diffusional constant-slip model. This new scaling law unifies a diverse class of slow seismic events and may lead to a better understanding of the plate subduction process and large earthquake generation.