Projective limits of probability spaces

The classical Kolmogorov theorem on the existence of stochastic process has been generalized in several directions following its abstract formulation by Bochner. In the first half of the paper a unified exposition of the key results of the existing work is given. The second half consists of some characterizations of the projective systems admitting projective limits and some applications. The latter include a generalization of a theorem of Tulcea on product measures involving conditional probabilities, which now need not be regular, and a characterization of the regular martingale of Chow and Snell, as a particular projective system admitting the projective limit. Comparisons with other work and some pertinent remarks are included at several places.