  We extend tropicalization and tropical compactification of subvarieties of
algebraic tori to subvarieties of spherical homogeneous spaces. Given a
tropical compactification of a subvariety, we show that the support of the
colored fan of the ambient spherical variety agrees with the tropicalization of
the subvariety. The proof is based on our equivariant version of the attening
by blow-up theorem. We provide many examples.
