Introduction; Classical foundations; Surface-transformations: Lie's first questions, finite-order Generalization, Infinite- order structure; Tranformation of families of surfaces: Lie's second question, Bianchi-Lie tranformation, Backlund transformations; Examples of Backlund transformations: invariance transformations; Transformations relating different differential equations; Tangent transformation groups: Finite-order tangent tranformations, tangent transformation groups of Sophus Lie, higher-order tangent transformation groups, infinite-order tangent transformations; Lie-Backlund tangent transformation Groups, Lie-Backlund equations; Application to differential equations: defining equations; Group theoretical nature of conservation laws, Lie via Lie-Backlund for ordinary differential equations, group theoretical equivalence of quantum-mechanical systems; Some applications of Backlund transformations: Nonlinear optics, solitons and the KdV equation, constants of the motion and conservation laws, weakly dispersive shallow-water waves in two space dimensions.
