Midwest Algebra, Geometry and their Interactions Conference MAGIC05 University of Notre Dame, Notre Dame October 7-11, 2005 Toroidalization of projective morphisms by Steven Dale CUTKOSKY, University of Missouri at Columbia Abstract: Suppose that f:X ® Y is a proper morphisms of varieties. A toroidalization of f is a sequence of blow ups of nonsingular centers X1 ® X and Y1 ® Y such that there is an induced morphism f1 : X1 ® Y1 which is locally formally isomorphic to a morphism of toric varieties. That is, f is formally a monomial mapping. We discuss our proof of toroidalization of morphisms of 3-folds. We further discuss some related invariants, and some of the difficulties in extending the proof to higher dimensions.