  A survey of some recent and important results which have to do with
integrable equations and their relationship with the theory of surfaces is
given. Some new results are also presented. The concept of the moving frame is
examined, and it is used in several subjects, which are discussed. Structure
equations are introduced in terms of differential forms. Forms are shown to be
very useful in relating geometry, equations and surfaces, which appear in many
sections. The topics of the chapters are different and separate, but joined
together by common themes and ideas. Several subjects which are not easy to
access are elaborated, such as Maurer-Cartan cocycles and recent results with
regard to generalizations of the Weierstrass-Enneper method for generating
constant mean curvature surfaces in three and higher dimensional Euclidean
spaces.
