We discuss the integration of Poisson brackets, motivated by our recent solution to the
integrability problem for general Lie brackets. We give the precise obstructions to
integrating Poisson manifolds, describing the integration as a symplectic quotient, in the
spirit of the Poisson sigma-model of Cattaneo and Felder. For regular Poisson manifolds we
express the obstructions in terms of variations of symplectic areas, improving on results
of Alcalde Cuesta and Hector. We apply our results (and our point of view) to decide about
the existence of complete symplectic realizations, to the integrability of submanifolds of
Poisson manifolds, and to the study of dual pairs, Morita equivalence and reduction.