Lagrangian fibrations arise naturally in the study of Liouville integrable systems and can be used to construct topological and symplectic invariants of such dynamical systems. As shown by Weinstein and Duistermaat amongst others, Lagrangian fibrations (and, more generally, foliations) are connected with affinely flat geometry, i.e. the differential geometry of those manifolds which admit a flat, torsion-free connection. In this talk, the problem of constructing Lagrangian bundles over a fixed manifold is discussed using affinely flat geometry; it is proved that the obstruction to constructing examples with non-trivial topological invariants is determined the radiance obstruction, an important cohomological invariant of affinely flat manifolds introduced by Goldman and Hirsch. Time permitting, I will illustrate how to extend this theory to the case with singularities.