Quasi, twisted, and all that... in Poisson geometry and Lie algebroid theory

Motivated by questions from quantum group and field theories, we review structures on manifolds that are weaker versions of Poisson structures, and variants of the notion of Lie algebroid. We give a simple definition of the Courant algebroids and introduce the notion of a deriving operator for the Courant bracket of the double of a proto-bialgebroid. We then describe and relate the various quasi-Poisson structures, which have appeared in the literature since 1991, and the twisted Poisson structures studied by Severa and Weinstein.