Poisson sigma models and deformation quantization

This is a review aimed at the physics audience on the relation between Pois son sigma models on surfaces with boundary and deformation quantization. Th ese models are topological open string theories. In the classical Hamiltoni an approach, we describe the reduced phase space and its structures (symple ctic groupoid), explaining in particular the classical origin of the noncom mutativity of the string endpoint coordinates. We also review the perturbat ive Lagrangian approach and its connection with Kontsevich's star product. Finally we comment on the relation between the two approaches.