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.