An introduction to constructive desingularization

These are introductional notes to resolution of singularities and Log principalization of ideals over fields of characteristic zero. We refer to A simplified proof of desingularization and applications', A. Bravo, S. Encinas and O. Villamayor, (to appear in {\em Revista Matem\'atica Iberoamericana}), for more details. The notes follow the algorithmic proof in A new proof of desingularization over fields of characteristic zero’, S. Encinas and O. Villamayor U., {\em Revista Matem'atica Iberoamericana}, 19 (2) (2003), which provides an important conceptual simplification over the original proofs of Hironaka. In fact Hironaka’s the notions of standard basis, Hilbert Samuel functions, and normal flatness are avoided.