The 1953 Beckman-Quarles theorem asserts that a function f from a Euclidean space of dimension at least two to itself with the property that d(f(x),f(y))=1 whenever d(x,y)=1 is an isometry. I'll discuss a conjectural Riemannian generalization of this theorem and some supporting results. Based on joint work with Meera Mainkar. Speaker(s): Ben Schmidt (MSU)