  We give a complete description and classification of locally homogeneous real
hypersurfaces in $\mathbb C^3$. Various groups of mathematicians have been
studying this problem in the last 25 years, and several significant classes of
hypersurfaces under consideration have been studied and classified. The final
results in the classification problem presented in this paper are obtained by
using the classification of abstract 5-dimensional real Lie algebras, and by
studying their representations by algebras of holomorphic vector fields in
complex 3-space. The complete list of pairwise inequivalent hypersurfaces that
we obtain contains 47 types of homogeneous hypersurfaces; some of the types are
1- or 2-parametric families, and each of the others is single hypersurface or a
finite family of hypersurface.
