We have investigated for the first time the propagation of the avalanche multiplication over the area of p-n junctions reverse biased above the breakdown voltage. The multiplication process spreads from the point where the avalanche is triggered to the whole junction area with a speed proportional to the final steady-state value of the avalanche current. The values of the propagation speed suggest that the phenomenon is due to diffusion of carriers assisted by avalanche multiplication. This effect strongly affects the rise of the avalanche current and turns out to limit the performance of single photon avalanche diodes.