  The Danielewski hypersurfaces are the hypersurfaces $X_{Q,n}$ in
$\mathbb{C}^3$ defined by an equation of the form $x^ny=Q(x,z)$ where $n\geq1$
and $Q(x,z)$ is a polynomial such that $Q(0,z)$ is of degree at least two. They
were studied by many authors during the last twenty years. In the present
article, we give their classification as algebraic varieties. We also give
their classification up to automorphism of the ambient space. As a corollary,
we obtain that every Danielewski hypersurface $X_{Q,n}$ with $n\geq2$ admits at
least two non-equivalent embeddings into $\mathbb{C}^3$.
