  In this paper we present the solution to a longstanding problem of
differential geometry: Lie's third theorem for Lie algebroids. We show that the
integrability problem is controlled by two computable obstructions. As
applications we derive, explain and improve the known integrability results, we
establish integrability by local Lie groupoids, we clarify the smoothness of
the Poisson sigma-model for Poisson manifolds, and we describe other
geometrical applications.
