  A set of necessary conditions for $C^1$ stability of noninvertible maps is
presented. It is proved that the conditions are sufficient for $C^1$ stability
in compact oriented manifolds of dimension two. An example given by F.Przytycki
in 1977 is shown to satisfy these conditions. It is the first example known of
a $C^1$ stable map (noninvertible and nonexpanding) in a manifold of dimension
two, while a wide class of examples are already known in every other dimension.
\end{abstract}
