  This paper contains a short and simplified proof of desingularization over
fields of characteristic zero, together with various applications to other
problems in algebraic geometry (among others, the study of the behavior of
desingularization of families of embedded schemes, and a formulation of
desingularization which is stronger than Hironaka's). Our proof avoids the use
of the Hilbert-Samuel function and Hironaka's notion of normal flatness: First
we define a procedure for principalization of ideals (i.e. a procedure to make
an ideal invertible), and then we show that desingularization of a closed
subscheme $X$ is achieved by using the procedure of principalization for the
ideal ${\mathcal I}(X)$ associated to the embedded scheme (X). The paper
intends to be an introduction to the subject, focused on the motivation of
ideas used in this new approach, and particularly on applications, some of
which do not follow from Hironaka's proof.
