  On July 26, 1995, at the University of California, Santa Cruz, a young Dutch
mathematician by the name Aise Johan de Jong made a revolution in the study of
the arithmetic, geometry and cohomology theory of varieties in positive or
mixed characteristic. The talk he delivered, first in a series of three
entitled "Dominating Varieties by Smooth Varieties", had a central theme: a
systematic application of fibrations by nodal curves. Among the hundreds of awe
struck members of the audience, participants of the American Mathematical
Society Summer Research Institute on Algebraic Geometry, many recognized the
great potential of Johan de Jong's ideas even for complex algebraic varieties,
and indeed soon more results along these lines began to form.
  This paper is an outgrowth of our course material prepared for the Working
Week on Resolution of Singularities, which was held during September 7-14, 1997
in Obergurgl, Tirol, Austria. As we did in the workshop, we intend to explain
Johan de Jong's results in some detail, and give some other results following
the same paradigm, as well as a few applications, both arithmetic and in
characteristic zero. We hope that the reader will come to share some of the
excitement we felt on that beautiful July day in Santa Cruz.
