This paper is a review of certain results related to the holomorphic continuation of functions given on the boundary of a bounded domain \(D\subset\mathbb C^n\), \(n> 1\), to this domain. The subject is far from new, but here the authors speak of functions with a one-dimensional holomorphic continuation property along complex lines and curves and also of boundary higher-dimensional variants of the Morera theorem. On the complex plane \(\mathbb C\) the results on functions with the one-dimensional holomorphic continuation property are trivial, and there are no boundary Morera theorems. Therefore, most of the results of the paper are essentially higher-dimensional. The authors consider global boundary analogs of the Morera theorem along complex and real planes, various results related to the Morera property along complex lines, different variants of the boundary Morera theorem in a ball, global analogs of the boundary Morera theorems along complex curves and local variants of the Morera theorems. Results on functions with the Morera property in classical domains are presented, and the Morera theorems for unbounded domains are given. The authors formulate certain unsolved problems also.