Affinely Homogeneous Real Hypersurfaces of $\mathbb{C}^2$

A complete affine classification of germs of affinely homogeneous real hypersurfaces of $\mathbb{C}^2$ is presented. The two main tools used in the classification are canonical local equations of manifolds and the theory of Lie algebras. The classification obtained in the paper is shown to be different from the well-known description of holomorphically homogeneous real hypersurfaces of $\mathbb{C}^2$ due to {É.} Cartan (1932).