Matroids are combinatorial objects designed to capture essential features of the notion of (linear) independence. They show up naturally in several contexts, including algebraic geometry, topology and optimisation. We will introduce an invariant of matroids called the h-vector that arises when studying the matroid in algebraic and topological contexts. We will explain an old conjecture of Stanley about these h-vectors, survey what is known and explain a new connection with discrete geometry. We will not assume previous knowledge of matroid theory.