Abstract: The genus integration of a Lie algebroid was introduced in joint work with Rui Fernandes as the quotient space of A-paths by A-homologies. We prove that such integration is the abelianization of the Weinstein groupoid, and we also show that the obstructions to smoothness of the genus integration are controlled by the extended monodromy groups. In this talk we will survey the integration problem of Lie algebroids and these new results, and we will present an ongoing work on the particular case of the prequantization Lie algebroid associated with a closed 2-form. In particular our results recover the prequantization condition as well as the usual description of principal circle bundles with connection via differential characters. Based on joint work with Rui Fernandes.