Multiplicity-Free Schubert Calculus Multiplicity-free algebraic geometry is the study of subvarieties $Y\subseteq X$ with the smallest invariants'' as witnessed by a multiplicity-free Chow ring decomposition of $[Y]\in A^{\star}(X)$ into a predetermined linear basis. This paper concerns the case of Richardson subvarieties of the Grassmannian in terms of the Schubert basis. We give a nonrecursive combinatorial classification of multiplicity-free Richardson varieties, i.e., we classify multiplicity-free products of Schubert classes. This answers a question of W. Fulton. Categories:14M15, 14M05, 05E99