A toric Lagrangian fibration is a Lagrangian fibration whose singular fibers are all of elliptic type. I will begin by explaining how such fibrations can be viewed as Hamiltonian spaces associated with symplectic torus bundles. I will then discuss a generalization to this class of fibrations of the Abreu–Guillemin–Donaldson theory of extremal Kähler metrics on toric symplectic manifolds. Integral affine geometry plays a central role in this generalization, as the Delzant polytope is replaced by a more general domain contained in an integral affine manifold. This talk is based on on-going work with Miguel Abreu and Maarten Mol.