Modifications of bundles, elliptic integrable systems, and related problems

We describe a construction of elliptic integrable systems based on bundles with nontrivial characteristic classes, especially attending to the bundle-modification procedure, which relates models corresponding to different characteristic classes. We discuss applications and related problems such as the Knizhnik-Zamolodchikov-Bernard equations, classical and quantum R-matrices, monopoles, spectral duality, Painlevé equations, and the classical-quantum correspondence. For an SL(N,?)-bundle on an elliptic curve with nontrivial characteristic classes, we obtain equations of isomonodromy deformations.