Kolmogorov and boundary problems of probability theory

This paper consists of two parts. The first part traces the history of the boundary problems in probability theory in which Kolmogorov took the most active part. A recently discovered new approach to the solution of boundary problems for random walks satisfying the Cramér condition is presented. This approach is more general, simple, and intuitive than the analytic method proposed by the author in the 1960s. Kolmogorov thought that the construction of such a more general alternative approach was quite desirable because, in his opinion, the purely analytic approach was not quite adequate in essence. The second part of the paper involves the main limit theorems in boundary problems for random walks not satisfying the Cramér condition. A number of recently obtained results are given.