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 reviewed and elaborated upon. These topics include Maurer-Cartan cocycles
and recent results with regard to generalizations of the Weierstrass-Enneper system forgenerating constant mean curvature surfaces in three and higher dimensional Euclidean
spaces.