Holomorphic Continuation of Functions Along a Fixed Direction (Survey)

In this article, we give an overview of the most significant and important results on holomorphic extensions of functions along a fixed direction. We discuss the following geometric questions of multidimensional complex analysis: holomorphic extension along a bundle of complex straight line, the Forelly theorem; holomorphic continuation of functions with thin singularities along a fixed direction; holomorphic continuation of functions along a family of analytic curves.