I. Introduction.- II. Integrable Hamiltonian systems on affine Poisson varieties.- III. Integrable Hamiltonian systems and symmetric products of curves.- IV. Interludium: the geometry of Abelian varieties.- V. Algebraic completely integrable Hamiltonian systems.- VI. The master systems.- VII. The Garnier and Hénon-Heiles potentials and the Toda lattice.- References.