Darboux Transformations in Integrable Systems : theory and their applications to geometry The Darboux transformation approach is one of the most effective methods for constructing explicit solutions of partial differential equations which are called integrable systems and play important roles in mechanics physics and differential geometry. This book presents the Darboux transformations in matrix form and provides purely algebraic algorithms for constructing the explicit solutions. A basis for using symbolic computations to obtain the explicit exact solutions for many integrable systems is established. Moreover the behavior of simple and multi solutions even in multi dimensional cases can be elucidated clearly. The method covers a series of important equations such as various kinds of AKNS systems in R1+n harmonic maps from 2 dimensional manifolds self dual Yang Mills fields and the generalizations to higher dimensional case theory of line congruences in three dimensions or higher dimensional space etc. All these cases are explained in detail. This book contains many results that were obtained by the authors in the past few years.