A remarkable theorem due to Khovanskii asserts that for any finite subset A of an abelian semigroup, the cardinality of the h-fold sumset hA grows like a polynomial for all sufficiently large h. However, neither the polynomial nor what sufficiently large means are understood in general. In joint work with Michael Curran (Oxford), we obtain an effective version of Khovanskii's theorem for any subset of $\mathbb{Z}^d$ whose convex hull is a simplex; previously such results were only available for d=1. Our approach also gives information about the structure of hA, answering a recent question posed by Granville and Shakan. The talk will be broadly accessible; interested mathematicians from any field are encouraged to attend.