Integrability of Lie brackets

In this paper we present the solution to a longstanding problem of dierential geometry Lies 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 sigmamodel 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 Lies third theorem given by Duistermaat and Kol