Let $��: P\to B$ be a locally trivial fiber bundle over a connected CW complex $B$ with fiber equal to the closed symplectic manifold $(M,\om)$. Then $��$ is said to be a symplectic fiber bundle if its structural group is the group of symplectomorphisms $\Symp(M,\om)$, and is called Hamiltonian if this group may be reduced to the group $\Ham(M,\om)$ of Hamiltonian symplectomorphisms. In this paper, building on prior work by Seidel and Lalonde, McDuff and Polterovich, we show that these bundles have interesting cohomological properties. In particular, for many bases $B$ (for example when $B$ is a sphere, a coadjoint orbit or a product of complex projective spaces) the rational cohomology of $P$ is the tensor product of the cohomology of $B$ with that of $M$. As a consequence the natural action of the rational homology $H_k(\Ham(M))$ on $H_*(M)$ is trivial for all $M$ and all $k > 0$. Added: The erratum makes a small change to Theorem 1.1 concerning the characterization of Hamiltonian bundles.

40 pages, Latex. Erratum added. Comments on previous version: shortened, section on 4-dimensional bases omitted, minor corrections and extra discussion of the homotopy aspects of the problem