In general, one cannot expect that every point of the intersection of the tropicalizations of two varieties can be lifted
to the intersection of the varieties themselves. A result of Bogart, Jensen, Speyer, Sturmfels and Thomas asserts that one always has such a lifting if the tropicalizations intersect suitably transversely. In joint work with Sam Payne, we generalize and strengthen their result in three directions: we weaken the transversality hypothesis, we work with intersections in more general ambient varieties, and we also treat intersection multiplicities. We hope that these results can be applied to moduli spaces to prove correspondence theorems relating enumerative algebraic geometry to enumerative tropical geometry.