  This is the expanded text of a series of CIME lectures. We present an
algebro-geometric approach to integrable systems, starting with those which can
be described in terms of spectral curves. The prototype is Hitchin's system on
the cotangent bundle of the moduli space of stable bundles on a curve. A
variant involving meromorphic Higgs bundles specializes to many familiar
systems of mathematics and mechanics, such as the geodesic flow on an ellipsoid
and the elliptic solitons. We then describe some systems in which the spectral
curve is replaced by various higher dimensional analogues: a spectral cover of
an arbitrary variety, a Lagrangian subvariety in an algebraically symplectic
manifold, or a Calabi-Yau manifold. One peculiar feature of the CY system is
that it is integrable analytically, but not algebraically: the Liouville tori
(on which the system is linearized) are the intermediate Jacobians of a family
of Calabi-Yau manifolds. Most of the results concerning these three types of
non-curve-based systems are quite recent. Some of them, as well as the
compatibility between spectral systems and the KP hierarchy, are new, while
other parts of the story are scattered through several recent preprints. As
best we could, we tried to maintain the survey style of this article, starting
with some basic notions in the field and building gradually to the recent
developments.
