Probability distributions on topological groups

The theory of probability distributions on algebraic structures is now rapidly developping. A review of many results in this field is to be found in Grenander's book [1]. But simultaneously with and after Grenander's book there appeared a number of papers that give in some aspects much more profound information than the book. Besides some earlier works are also left by U. Grenander without due consideration.
The present paper is a review concerned only with the case of topological groups. Some new results are also included. This review and corresponding chapters of Grenander's book can be regarded as mutually complementary.