  The main goal of the present paper is two-fold. First we extend the theory of
toroidal embeddings introduced by Kempf, Knudsen, Mumford and Saint-Donat to
the class of toroidal varieties with stratifications (which is the main body of
the paper). Second we give a proof of the following weak factorization theorem
as an application and illustration of the theory: A birational map between
complete nonsingular varieties over an algebraically closed field K of
characteristic zero is a composite of blow ups and blow downs with smooth
centers.
  Another proof of the weak factorization theorem appeared in a joint paper
with Abramovich, Karu and Matsuki (math.AG/9904135) In that paper the theorem
is stated and proven in general for proper algebraic and analytic spaces.
