  We introduce a notion of stable spherical variety which includes the
spherical varieties under a reductive group $G$ and their flat equivariant
degenerations. Given any projective space $\bP$ where $G$ acts linearly, we
construct a moduli space for stable spherical varieties over $\bP$, that is,
pairs $(X,f)$, where $X$ is a stable spherical variety and $f : X \to \bP$ is a
finite equivariant morphism. This space is projective, and its irreducible
components are rational. It generalizes the moduli space of pairs $(X,D)$,
where $X$ is a stable toric variety and $D$ is an effective ample Cartier
divisor on $X$ which contains no orbit. The equivariant automorphism group of
$\bP$ acts on our moduli space; the spherical varieties over $\bP$ and their
stable limits form only finitely many orbits. A variant of this moduli space
gives another view to the compactifications of quotients of thin Schubert cells
constructed by Kapranov and Lafforgue.
