Toroidal varieties and their morphisms were introduced and developed by Kempf, Knudsen, Mumford and Saint-Donat in 1973, in order to prove the Semistable Reduction Theorem (simplifying a singular fiber of a morphism of complex varieties). Since then they have been successfully applied to various problems concerning the resolution of singularities and factorization of morphisms. Among them is the Problem of Toroidalization which is concerned with obtaining a toroidal lifting of a given morphism f: X to Y of algebraic varieties by performing sequences of blow-ups of non-singular sub-varieties above X and Y. The purpose of this talk is to introduce this problem and the results obtained so far.