  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.
