  We give several characterizations of stable intersections of tropical cycles
and establish their fundamental properties. We prove that the stable
intersection of two tropical varieties is the tropicalization of the
intersection of the classical varieties after a generic rescaling. A proof of
Bernstein's theorem follows from this. We prove that the tropical intersection
ring of tropical cycle fans is isomorphic to McMullen's polytope algebra. It
follows that every tropical cycle fan is a linear combination of pure powers of
tropical hypersurfaces, which are always realizable. We prove that every stable
intersection of constant coefficient tropical varieties defined by prime ideals
is connected through codimension one. We also give an example of a realizable
tropical variety that is connected through codimension one but whose stable
intersection with a hyperplane is not.
