Following a brief review of relevant theoretical and empirical spatial results, a theory of space-time rainfall, applicable to fields advecting without deformation of the coordinates, is presented and tested: In this theory, spatial rainfall fields are constructed from discrete multiplicative cascades of independent and identically distributed (iid) random variables called generators. An extension to space-time assumes that these generators are iid stochastic processes indexed by time. This construction preserves the spatial structure of the cascades, while enabling it to evolve in response to a nonstationary large-scale forcing, which is specified externally. The construction causes the time and space dimensions to have fundamentally different stochastic structures. The time dimension of the process has an evolutionary behavior that distinguishes between past and future, while the spatial dimensions have an isotropic stochastic structure. This anisotropy between time and space leads to the prediction of the breakdown of G. I. Taylor's hypothesis of fluid turbulence after a short time, as is observed empirically. General, nonparametric, predictions of the theory regarding the spatial scaling properties of two-point temporal cross moments are developed and applied to a tracked rainfall field in a case study. These include the prediction of the empirically observed increase of correlation times as resolution decreases and the scaling of temporal cross moments, a new finding suggested by this theory. ¿ American Geophysical Union 1996