  We give a construction of homotopy algebras based on ``higher derived
brackets''. More precisely, the data include a Lie superalgebra with a
projector on an Abelian subalgebra satisfying a certain axiom, and an odd
element $\Delta$. Given this, we introduce an infinite sequence of higher
brackets on the image of the projector, and explicitly calculate their
Jacobiators in terms of $\Delta^2$. This allows to control higher Jacobi
identities in terms of the ``order'' of $\Delta^2$. Examples include Stasheff's
strongly homotopy Lie algebras and variants of homotopy Batalin--Vilkovisky
algebras. There is a generalization with $\D$ replaced by an arbitrary odd
derivation. We discuss applications and links with other constructions.
