A hyperbolic manifold is a Riemannian manifold of constant curvature -1. Compact (more generally, complete finite-volume) hyperbolic manifolds exist in every dimension n, but our understanding of their topology and geometry is still very limited when n>3. One of the most striking aspects of 3-manifolds is that many of them are total spaces of bundles over the circle, so their topology is determined by the monodromy, which is in turn beautifully described by Thurston's theory of diffeomorphisms of surfaces. We will show here that a similar picture arises for some hyperbolic 5-manifolds. This is joint work with Italiano and Migliorini.