Extension of Holomorphic and Pluriharmonic Functions with Thin Singularities on Parallel Sections

The paper is of survey character. We present and discuss recent results concerning the extension of functions that admit holomorphic or plurisubharmonic extension in a fixed direction. These results are closely related to Hartogs' fundamental theorem, which states that if a function $f(z)$, $z = (z_1,z_2,\dots ,z_n)$, is holomorphic in a domain $D\subset \mathbb C^n$ in each variable $z_j$, then it is holomorphic in $D$ in the $n$-variable sense.