  We survey the many instances of derived bracket construction in differential
geometry, Lie algebroid and Courant algebroid theories, and their properties.
We recall and compare the constructions of Buttin and Vinogradov, and we prove
that the Vinogradov bracket is the skew-symmetrization of a derived bracket.
Odd (resp., even) Poisson brackets on supermanifolds are derived brackets of
canonical even (resp., odd) Poisson brackets on their cotangent bundle (resp.,
parity-reversed cotangent bundle). Lie algebras have analogous properties, and
the theory of Lie algebroids unifies the results valid for manifolds on the one
hand, and for Lie algebras on the other. We outline the role of derived
brackets in the theory of "Poisson structures with background".
