Seiberg-Witten geometry of mass deformed N=2 superconformal ADE quiver gauge
theories in four dimensions is determined. We solve the limit shape equations
derived from the gauge theory and identify the space M of vacua of the theory
with the moduli space of the genus zero holomorphic (quasi)maps to the moduli
space of holomorphic G-bundles on a (possibly degenerate) elliptic curve
defined in terms of the microscopic gauge couplings, for the corresponding
simple ADE Lie group G. The integrable systems underlying, or, rather,
overlooking the special geometry of M are identified. The moduli spaces of
framed G-instantons on R^2xT^2, of G-monopoles with singularities on R^2xS^1,
the Hitchin systems on curves with punctures, as well as various spin chains
play an important role in our story. We also comment on the higher dimensional
theories. In the companion paper the quantum integrable systems and their
connections to the representation theory of quantum affine algebras will be
discussedComment: 197 page