(joint work with R.L. Fernandes, preprint, 2001)
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. Our approach also puts into a new perspective the work of Cattaneo and Felder for the special case of Poisson manifolds and the "new" proof of Lie's third theorem given by Duistermaat and Kolk.