The moduli space of vector bundles on a variety X which admits an elliptic fibration f : X → B (with a section σ : B → X) can be described in terms of data on the base B via the Fourier-Mukai transform, or the spectral construction. This is an ”elliptic” version of the various integrable systems of meromorphic Higgs fields on B: instead of Higgs fields with values in the canonical bundle (or any other vector bundle) on B, one considers Higgs fields with values in the family f of elliptic curves. The extension of this result to the case where f : X → B is a genus 1 fibration (having no section σ : B → X) leads to some surprising new features, including the appearance of gerbes, or non- commutative structures, on X. In a sense, the non-commutativity of such a structure is dual to the non-existence of a section. I will formulate a still partly conjectural duality of derived categories on these gerbes, and will show how it incorporates the Fourier-Mukai transform and other known results. Mirror symmetry between Calabi-Yau threefolds is conjectured (by Strominger, Yau and Zaslow) to be realized geometrically by a duality quite similar to Fourier-Mukai, but involving special Lagrangian tori instead of elliptic curves. The CY analogue of our results suggests a modification and strengthening of the SYZ conjecture. In particular, there should be a real integrable system on the stringy moduli space of Calabi-Y (i.e. the moduli of the complex structure, Kahler structure, and B-field) on which mirror symmetry acts. (Joint with Tony Pantev).