The author discusses in detail six papers of Kolmogorov on approximation theory and their influence on this theory. Table of contents: §1. Papers of A. N. Kolmogorov on approximation theory. §2. Exact bounds of approximation on classes. §3. Diameters. §4. Inequalities for derivatives. §5. Diameters of finite-dimensional sets. §6. Criterion of the shortest elements. §7. Proofs.