Grenander's text, a systematic account of the theory of probability for certain sample spaces with algebraic structure, covers semi-groups, groups, linear vector spaces, and algebra. It states and studies fundamental probabilistic problems for these spaces, focusing on concrete results rather than on general and complete answers. An introductory chapter offers historical background on the study of probabilities. Subsequent chapters explore stochastic semi-groups, compact and commutative stochastic groups, stochastic Lie groups, and locally compact stochastic groups. Additional topics include stochastic linear spaces and stochastic algebras. The techniques of Fourier analysis receive particular attention, serving as the main analytical tools in obtaining limit theorems for convolutions of probability distributions. Theoretical developments are accompanied by a number of examples, some of them of considerable scope.