1. Algebraic Foundations: valuations, rings of valuations, the algebraic definition of the Riemann-Zariski space. Examples, especiall attention to the case of valuations of the field k(x, y) in two variables. Generating series of Suite a valuation of k(x, y). Definition of regular and singular points.
2. Geometric foundations: singularities, blow-up (along an arbitrary ideal), an introduction to the problem of resolution of singularities. Examples, with particular attention to the cases of singularities of plane curves (and their relationship with the generating series).
3. Geometric definition of the Riemann-Zariski space; demonstration of the equivalence of two definitions, the algebraic and the geometric.
4. Demonstration the compactness of the Zariski-Riemann space. Statement of the of Local Uniformisation Theorem and Zariski's approach to the problem of resolution of singularities though the compactification of Riemann-Zarski's space.
5. If time permits, we shall study the generalization of generating series and resolution of singularities of plane curves in the case of resolution of singularities in any dimension and arbitrary characteristic.