In this talk, we discuss generalized geometric reduction procedures which lead to Poisson (and related) structures on the reduced space but in which the starting "reduction data" is of more general type. The idea behind it is based on the characterization of the relevant geometric structures as Lie algebroids equipped with compatible 2-forms. The main results involve quotients of such Lie algebroids, their corresponding Lie theoretic integration to Lie groupoids endowed with multiplicative 2-forms, and the infinitesimal characterization of non-degeneracy conditions. Finally, we discuss several concrete applications, recovering particular cases and showing how these results can be applied in various problems. This talk is based on joint work with C. Ortiz.