  We give a survey of the following six closely related topics: (i) a general
method for constructing a soliton hierarchy from a splitting of a loop algebra
into positive and negative subalgebras, together with a sequence of commuting
positive elements, (ii) a method---based on (i)---for constructing soliton
hierarchies from a symmetric space, (iii) the dressing action of the negative
loop subgroup on the space of solutions of the related soliton equation, (iv)
classical B\"acklund, Christoffel, Lie, and Ribaucour transformations for
surfaces in three-space and their relation to dressing actions, (v) methods for
constructing a Lax pair for the Gauss-Codazzi Equation of certain submanifolds
that admit Lie transforms, (vi) how soliton theory can be used to generalize
classical soliton surfaces to submanifolds of higher dimension and
co-dimension.
