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A survey of some recent and important results which have to do withintegrable equations and their relationship with the theory of surfaces isgiven. Some new results are also presented. The concept of the moving frame isexamined, and it is used in several subjects, which are discussed. Structureequations are introduced in terms of differential forms. Forms are shown to bevery useful in relating geometry, equations and surfaces, which appear in manysections. The topics of the chapters are different and separate, but joinedtogether by common themes and ideas. Several subjects which are not easy toaccess are elaborated, such as Maurer-Cartan cocycles and recent results withregard to generalizations of the Weierstrass-Enneper method for generatingconstant mean curvature surfaces in three and higher dimensional Euclideanspaces.

The interrelationship of integrable equations, differential geometry and the geometry of their associated surfaces