The deep connections that exist between the classical differential geometry of surfaces and modern soliton theory are by now well established. Thus, Bäcklund transformations, together with Darboux-type transformations in the form of the Levy transformation and the so-called Fundamental Transformation of differential geometry, have proved to be important tools in the generation of solutions to the nonlinear equations of soliton theory. Eisenhart, in the preface to his monograph Transformations of Surfaces published in 1922, asserted that

Thus, distinguished geometers such as Bianchi, Calapso, Darboux, Demoulin, Guichard, Jonas, Ribaucour, and Weingarten all conducted detailed investigations into various privileged classes of surfaces that admit such transformations.

It is with the class of surfaces that admit invariance under Bäcklund-Darboux transformations that the present monograph is concerned. Invariance under a Bäcklund transformation turns out to be a generic property of all solitonic equations. In the geometric context of this monograph, solitonic equations are seen to arise out of the nonlinear Gauss-Mainardi-Codazzi equations for various types of surfaces that admit invariance under Bäcklund-Darboux transformations. The linear Gauss-Weingarten equations for such surfaces provide, on injection of a Bäcklund parameter, linear representations for the underlying nonlinear soliton equations.