The CR Hartogs theorem refers here to a property which may be true for domains in $\mathbb{C}^n$. Given a domain $D \subset \mathbb{C}^n$ with smooth boundary (usually real analytic), we say that the CR Hartogs theorem holds if for every function $f$ defined on $\partial D$ which has holomorphic extension to $D \cap L$ from $\partial D \cap L$, for every complex line parallel to a coordinate axis, $f$ extends as a fully holomorphic function to $D$. This theorem fails for the ball, but holds for rather generic strictly convex domains. In this talk I will show some generalizations in different directions.

1) A theorem for a domain which has nontrivial topology

2) A theorem for a domain which is a perturbation of an egg domain.

3) A theorem for meromorphic functions

4) A theorem for domains which cover domains in $\mathbb{C}^n$. There remains a natural question about identifying projective varieties by applying a local CR Hartogs theorem.