A growing body of observations worldwide has documented fault slip transients that radiate little or no seismic energy. The mechanisms that govern these slow slip events (SSEs) and their wide range of depths, slip rates, durations, stress drops and recurrence intervals remain poorly known. Here we show that slow slip can be explained by a transition from rate-weakening frictional sliding at low slip rates towards rate-neutral or rate-strengthening behaviour at higher slip rates, as has been observed experimentally. We use numerical simulations to illustrate that this rate-dependent transition quantitatively explains the experimental data for natural fault rocks representative of materials in the source regions of SSEs. With a standard constant-parameter rate-and-state friction law, SSEs arise only near the threshold for slip instability. The inclusion of velocity-dependent friction parameters substantially broadens the range of conditions for slow slip occurrence, and produces a wide range of event characteristics, which include stress drop, duration and recurrence, as observed in nature. Upscaled numerical simulations that incorporate parameters consistent with laboratory measurements can reproduce geodetic observations of repeating SSEs on tectonic faults. We conclude that slip-rate-dependent friction explains the ubiquitous occurrence of SSEs in a broad range of geological environments.