**Abstract:** Semistable reduction is a relative generalization of the classical problem of resolution of singularities of varieties; the goal is, given a surjective morphism $f : X \to B$ of varieties in characteristic 0, to change $f$ so that it is "as nice as possible". The problem goes back to at least Kempf, Knudsen, Mumford, and Saint-Donat (1973), who proved a strongest possible version when $B$ is a curve. The key ingredient in the proof is the following combinatorial result: Given any $d$-dimensional polytope $P$ with vertices in $\mathbb{Z}^d$, there is a dilation of $P$ which can be triangulated into simplices each with vertices in $\mathbb{Z}^d$ and volume $1/d!$. In 2000, Abramovich and Karu proved, for any base $B$, that $f$ can be made into a weakly semistable morphism $f' : X' \to B'$. They conjectured further that $f'$ can be made semistable, which amounts to making $X'$ smooth. They explained why this is the best resolution of $f$ one might hope for. In this talk I will outline a proof of this conjecture. They key ingredient is a relative generalization of the above combinatorial result of KKMS. I will also discuss some other consequences in combinatorics of our constructions. This is joint work with Karim Adiprasito and Michael Temkin.