The subject of this book is the hierarchies of integrable equations connected with the one-component and multi component loop groups. There are many publications on this subject, and it is rather well defined. Thus, the author would like t.o explain why he has taken the risk of revisiting the subject. The Sato Grassmannian approach, and other approaches standard in this context, reveal deep mathematical structures in the base of the integrable hi erarchies. These approaches concentrate mostly on the algebraic picture, and they use a language suitable for applications to quantum field theory....