This is a brief outline of results represented in the Ph.D. dissertation defended by the author in the Department of Mathematics at Moscow State University (MSU), Russia, in 1973. The complete manuscript of the dissertation is not published. Dissertation “Limit Theorems for Random Walks on Differential Manifolds” is dedicated to the generalization of classical limit theorems of Probability Theory, known as Central Limit Theorem, to the case when sequences of ‘random variables’ represent random walks on differentiable manifolds or generated by products of random elements of noncommutative locally compact groups Lie. Classical results of Probability Theory related to the summation of random variables provide a description of the family of limit laws for distribution of sums of uniformly distributed infinitesimal independent terms and establish conditions for convergence of the sums to the laws of the family. Problems related to probability distributions for sums of “small “ random variables were generalized in two major directions: first, it had been stated a more general problem to move from discrete random walks to Markov processes with continuous time (A.N. Kolmogorov [1], A. Y. Khintchin {2], A.V. Skorokhod [3]); second, the studies shifted to the situation in which the random variables are taking their values on sets of more general mathematical nature than n-dimensional Euclidean space, for example, differentiable manifolds and Lie groups (A. N. Kolmogorov [4], K. Ito [5], G.A. Hunt [6], D. When [7]. In our study, we consider sequences of random walks (i.e. random processes with discrete time) of both Markov and non-Markov character on locally-compact differentiable manifolds, and the limit transition from the walks to Markov random processes with continuous time so that the “steps” of walks are asymptotically uniformly small, and the number of “steps” goes to infinity. The mentioned Markov processes (diffusion and stochastically continuous without the second type of discontinuities) are introduced “constructively”, which provides the information about the properties of transition probabilities, necessary for proofs of the corresponding limit theorems. As a result, we formulate conditions of convergence, analogous to the classical theory. Limit theorems for probability distributions of products of asymptotically uniformly small random elements on Lie groups are proved as an application of the corresponding statements for manifolds. We assume no conditions related to the commutative property of measure convolutions. [Truncated to < 400 words by viXra Admin]