By Kodaira's work on the classification of compact complex surfaces, we know that there are three classes of surfaces with trivial canonical bundles: complex tori, K3 surfaces, and (primary) Kodaira surfaces. There are examples of all of these admitting elliptic fibrations, and Kodaira surfaces can even be described as `torsors' over elliptically-fibred tori.

A natural generalization of these surfaces in higher dimensions are holomorphic symplectic manifolds, and the generalization of an elliptic fibration is a Lagrangian fibration X->B, whose general fibre is a complex torus. In this talk we will describe the theory of `torsors' over these Lagrangian fibrations, showing how it connects various known examples of holomorphic symplectic manifolds.