Distributions of time between consecutive earthquakes verify an approximately universal scaling law for stationary seismicity. The shape of these distributions is shown to arise as a mixture of one distribution for short-distance events and an exponential distribution for far-off events, the distinction from short and long distances being relative to the size of the region studied. The distributions of consecutive distances show a double power law decay and verify an approximate scaling law which guarantees the simultaneous fulfillment of the scaling laws for time. The interplay between space and time can be seen as well by looking at the distribution of distances for a fixed time separation. These results suggest that seismicity can be understood as a series of intertwined independent continuous-time random walks, with power law-distributed waiting times and Lévy-flight jumps. However, a simple model based on these ideas does not capture the invariance of seismicity under renormalization. © 2007 Blackwell Publishing Ltd.