  We prove that if $M$ and $M'$ are algebraic hypersurfaces in $ C^ N$, i.e.
both defined by the vanishing of real polynomials, then any sufficiently smooth
CR mapping with Jacobian not identically zero extends holomorphically provided
the hypersurfaces are holomorphically nondegenerate . Conversely, we prove that
holomorphic nondegeneracy is necessary for this property of CR mappings to
hold. For the case of unequal dimensions, we also prove that if $M$ is an
algebraic hypersurface in $ C^N$ which does not contain any complex variety of
positive codimension and $M'$ is the sphere in $ C^{N+1 }$ , then extendability
holds for all CR mappings with certain minimal a priori regularity.
  Theorem A. Let $M$ and $M'$ be two algebraic hypersurfaces in $C^N$ and
assume that $M$ is connected and holomorphically nondegenerate. If $H$ is a
smooth CR mapping from $M$ to $M'$ with $ Jac H \not\equiv 0$, where $Jac H$ is
the Jacobian determinant of $H$, then $H$ extends holomorphically in an open
neighborhood of $M$ in $ C^N$.
  A recent example given by Ebenfelt shows that the conclusion of Theorem A
need not hold if $M$ is real analytic, but not algebraic.
  Theorem B. Let $M$ be a connected real analytic hypersurface in $C^N$ which
is holomorphically degenerate at some point $p_1$. Let $p_0 \in M$ and suppose
there exists a germ at $p_0$ of a smooth CR function on $M$ which does not
extend holomorphically to any full neighborhood of $p_0$ in $C^N$. Then there
exists a germ at $p_0$ of a smooth CR diffeomorphism from $M$ into itself,
fixing $p_0$, which does not extend holomorphically to any neighborhood of
$p_0$ in $C^N$.
  Theorem C. Let $M\subset C^N$ be an algebraic hypersurface. Assume that there
is no nontrivial complex analytic variety contained in $M$ through $p_0 \in M$,
and let $m=m_{p_0}$ be the D'Angelo type. If $H: M \to S^{2N+1}\subset C^{N+1}$
is a CR map of class $C^m$, where $S^{2N+1}$ denotes the boundary of the unit
ball in $C^{N+1 }$, then $H$ admits a holomorphic extension in a neighborhood
of $p_0$.
