In this paper we study associative algebras with a Poisson algebra structure on the center acting by derivations on the rest of the algebra. These structures, which we call Poisson fibred algebras, appear in the study of quantum groups at roots of 1 and related algebras, as well as in the representation theory of affine Lie algebras at the critical level. Poisson fibred algebras lead to a generalization of Poisson geometry, which we develop in the paper. We also take up the general study of noncommutative spaces which are close to enough commutative ones so that they contain enough points to have interesting commutative geometry. One of the most striking uses of our noncommutative spaces is the quantum Borel-Weil-Bott Theorem for quantum sl_q (2) at a root of unity, which comes as a calculation of the cohomology of actual sheaves on actual topological spaces.

21 pages, LaTeX