Holomorphically homogeneous real hypersurfaces in $ \mathbb{C}^3$

We discuss a complete solution of the problem on the local description and classification of holomorphically homogeneous real hypersurfaces in $ \mathbb{C}^3$. Large families of such manifolds have been studied by several groups of mathematicians using various approaches in the last 25 years. The final results in this problem have been obtained by the present author with the use of the classification of abstract five-dimensional real Lie algebras and the technique of their holomorphic representations in complex 3-space.
The complete list of pairwise nonequivalent holomorphically homogeneous hypersurfaces in our classification contains forty-seven types of such manifolds, including standalone hypersurfaces as well as one- and two-parameter families of hypersurfaces.