  The motivation of this work is to construct an analog of compactified moduli
of abelian varieties and toric pairs in the case of non-commutative algebraic
group G. We introduce a class of "stable reductive varieties" which contain
connected reductive groups and their equivariant compactifications, and is
closed under flat reduced degenerations. We classify them all, describe their
degenerations, and establish a connection between these varieties and
"reductive semigroups" which we also define. Finally, we construct a Hilbert
scheme of embedded G-varieties by applying and generalizing a construction of
Haiman and Sturmfels.
  The second version adds some cosmetic changes.
