The problem of biholomorphic classification of complex manifolds has been one of the central problems in Complex Analysis. Biholomorphic transformations are a very active and productive area of research. This article presents several directions in this work. After the general introduction of the subject, we examine related topics, which by themselves are highly interesting: invariant metrics, the extension of biholomorphic transformations to the boundary, automorphism groups of complex manifolds, fixed points, and determining sets of holomorphic self-maps of a domain. In the last section, we present some results on "approximate" biholomorphisms. © 2022 Taylor & Francis Group, LLC.