  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.
