Theoretical attempts to model space–time rainfall during the last 15 years have evolved along two separate lines. The first group of approaches is based on an assumed hierarchy of scales in spatial rainfall as noted by numerous empirical interpretations of remotely sensed observations. The second group of approaches rests on the assumption of an invariance property in statistical distributions of spatial rainfall called self-similarity, or simple scaling. The current theoretical developments involve a common modification of each of these assumptions. It is based on the notion of spatial random mass distribution or random measures generated by random cascades. The current mathematical foundations of the theory of random cascades involve the study of their ensemble properties and their sample-average properties in space. Recent results have shown that these two are not the same because the spatial law of large numbers does not hold for random cascades due to strong spatial correlations. Two key res... Abstract Theoretical attempts to model space–time rainfall during the last 15 years have evolved along two separate lines. The first group of approaches is based on an assumed hierarchy of scales in spatial rainfall as noted by numerous empirical interpretations of remotely sensed observations. The second group of approaches rests on the assumption of an invariance property in statistical distributions of spatial rainfall called self-similarity, or simple scaling. The current theoretical developments involve a common modification of each of these assumptions. It is based on the notion of spatial random mass distribution or random measures generated by random cascades. The current mathematical foundations of the theory of random cascades involve the study of their ensemble properties and their sample-average properties in space. Recent results have shown that these two are not the same because the spatial law of large numbers does not hold for random cascades due to strong spatial correlations. Two key res...