  We develop a classification theory for real-analytic hypersurfaces in
$\mathbb C^2$ in the case when the hypersurface is of {\em infinite type} at
the reference point. This is the remaining, not yet understood case in $\mathbb
C^2$ in the {\it Probl\`eme local}, formulated by H.\,Poincar\'e in 1907 and
asking for a complete biholomorphic classification of real hypersurfaces in
complex space. One novel aspect of our results, appearing in this revised
version, is a notion of {\em smooth normal forms} for real-analytic
hypersurfaces. We rely fundamentally on the recently developed CR -- DS
technique in CR-geometry.
