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One of the main motivation for studying probability theory on vector spaces is the possibility of applying the general theory to stochastic processes. In this paper we report the recent progress in the study of stable processes. It is true that the theory of stable measures on vector spaces and the theory of stable processes are developed rather separately. However, the aim of this survey is to show that there are many similarities and counnon points between them. It is our hope that this will stimulate more interactions and will help in understanding better the fascinating" stable world" so different from the Gaussian one. Why is this important?

We believe stable distributions and stable processes do provide useful models for many phenomena observed in diverse fie Ids. The most commonly met classical probability distribution functions in the analysis of physical processes are Gaussian or have exponential tails. In recent …