We consider the problem of constructing Poisson brackets on smooth manifolds
\(M\) with prescribed Casimir functions. If \(M\) is of even dimension, we achieve
our construction by considering a suitable almost symplectic structure on \(M\),
while, in the case where \(M\) is of odd dimension, our objective is achieved by
using a convenient almost cosymplectic structure. Several examples and
applications are presented.