The present thesis consists of three parts. In the first part we give a brief historical outline of the most important results that have been obtained in the theory of axially symmetric gravitational fields. Parts II and III are devoted to two contributions by the writer to this branch of the general theory of relativity. A bibliography of papers dealing with the subject is given at the end. This bibliography has been arranged in chronological order, and references to it are made by quoting the authors name followed by the year of publication.