Integrable Systems in the realm of Algebraic Geometry Introduction Integrable Hamiltonian systems on affine Poisson varietie: Affine Poisson varieties and their morphisms; Integrable Hamiltonian systems and their morphisms; Integrable Hamiltonian systems on other spaces Integrable Hamiltonian systems and symmetric products of curves: The systems and their integrability; The geometry of the level manifolds Interludium: the geometry of Abelian varieties: Divisors and line bundles; Abelian varieties; Jacobi varieties; Abelian surfaces of type (1 4) Algebraic completely integrable Hamiltonian systems: A.c.i. systems; Painlev analysis for a.c.i. systems; The linearization of two dimensional a.c.i. systems; Lax equations The Mumford systems: Genesis; Multi Hamiltonian structure and symmetries; The odd and the even Mumford systems; The general case Two dimensional a.c.i. systems and applications: The genus two Mumford systems; Application: generalized Kummersurfaces; The Garnier potential; An integrable geodesic flow on SO(4);...