Scale invariance, either in space or in time, has been shown in many papers to characterize earthquake distributions. Unfortunately, little work has been dedicated to looking at the general space-time scaling invariance of seismicity systems, even though a better understanding of how the two domains (spatial and temporal) link together could help the development of the stochastic dynamical modeling of earthquake populations. In this paper we report the observation of diffusion processes of temporally correlated seismic activity for three different data sets: a mine (Creighton Mine, Canada), the Long Valley Caldera in eastern California, and a 7-year period of recorded seismic activity in southern California. The observed subdiffusion processes are indicative of the general space-time scaling of the system, taking the form of a slow power law growth R(t)~tH of the mean distance R(t) between the main event and the temporally correlated afterevents occurring after a delay t. H is found on average to be small (0.1 for Creighton Mine, 0.22 for the Long Valley Caldera, and 0.22 for the southern California main events with magnitude≥1.5) but fluctuates significantly from one main event to the other: the diffusion is found to be intermittent (non-Gaussian) and multiscaling, and except for the Long Valley Caldera, a systematic correlation between the sizes of the main event and subsequent afterevents and the growth exponent H is observed. While classical viscous relaxation models (e.g., elastic listhosphere-plastic asthenosphere coupling, or fluid flow triggered by sudden changes in pore pressure) have been proposed to characterize this relaxation by homogeneous (i.e., nonintermittent) normal (H=0.5) diffusion processes, the direct implication of the reported results is that seismicity systems, at spatial scales from meters to hundreds of kilometers and small (microearthquakes in a mine) to intermediate magnitudes, relax spatiotemporally in a nonelastic way, revealing the stochastic space-time scale-invariant nature of such systems. Since these diffusion processes correspond to a loss of information with time on the location of the main event, they can be used to investigate the limits of predictability, at all spatial scales, of seismicity systems in terms of the spatiotemporal clustering of temporally correlated earthquakes. ¿ 2000 American Geophysical Union