Schubert calculus is the study of certain intersections of varieties in the flag manifold.  These intersections are /positive/ in the differential-geometric sense, but they also have “positivity properties” in several associated rings, notably equivariant cohomology and equivariant K-theory. We show how one can get new and old formulas for the structure constants in these rings using Bott-Samelson manifolds, a sequence of projective bundles lying over the flag manifold. Time permitting, we discuss the notion of positivity in the symplectic category, and explore how to extend these ideas in the more general context.