Jacobi brackets (a generalization of standard Poisson brackets in which
Leibniz's rule is replaced by a weaker condition) are extended to brackets
involving an arbitrary (even) number of functions. This new structure includes,
as a particular case, the recently introduced generalized Poisson structures.
The linear case on simple group manifolds is also studied and non-trivial
examples (different from those coming from generalized Poisson structures) of
this new construction are found by using the cohomology ring of the given
group.