Optimal transportation has enjoyed a mathematical renaissance over the last twenty-five years, weaving together threads from analysis, geometry, partial differential equations and dynamical systems. In the first lecture we introduce these mathematical developments, focusing particularly on duality theory, regularity questions, and connections to different notions of curvature in differential geometry and topology. The second lecture will concentrate on different applications to economic theory: stable matching problems, optimal decision making facing informational asymmetries, and equilibria for hierarchical multi-stage decision problems in multidimensional settings.