In joint work with Mak-Matessi-Zharkov, we prove the existence of Lagrangian torus fibrations on compact Calabi-Yau hypersurfaces in toric Fano manifolds given by a reflexive polytope. All singular fibers are half-dimensional, the discriminant in the base as well as the critical set are of real codimension two respectively and the fibration is smooth away from the discriminant. Consequently, the Arnold-Liouville-theorem identifies the smooth part with the cotangent bundle quotient of the base affine structure. The result is motivated by the Strominger-Yau-Zaslow conjecture which predicts the existence of these fibrations on Calabi-Yau manifolds near large complex structure limits such that mirror symmetry becomes T-duality. In this talk I will explain the main ideas that we have come up with over the years which solve the various hard problems that left the conjecture unsolved for several decades.