  Inspired by the work of Gross on topological Mirror Symmetry we construct
candidate Lagrangian torus fibration models for the 105 families of smooth Fano
threefolds. We prove, in the case the second Betti number is one, that the
total space of each fibration is homeomorphic to the expected Fano threefold,
and show that the numerical invariants coincide for all 105. Our construction
relies on a notion of toric degeneration for affine manifolds with
singularities, and the correspondence we obtain between polytopes and Fano
manifolds is compatible with that appearing in the work of
Coates-Corti-Kasprzyk et al. on Mirror Symmetry for Fano manifolds.
