We discuss a partly conjectural extension of the Fourier-Mukai transform, or the spectral construction, to several new situations, including fibrations without sections, non-commutative geometries, and higher dimensional complex or special Lagrangian torus fibers. The special Lagrangian case in particular casts new light on the Strominger-Yau Zaslow approach to mirror symmetry. It suggests the existence of an integrable system structure on the moduli space of Calabi-Yaus (with B- fields) on which mirror symmetry acts.