The appearance of infinite parameter symmetry transformations in particle theories is described. This invariance forms an affine algebra, a class of Kac-Moody Lie algebras, whose representations can be given by the dual string model. The symmetries exist in the two-dimensional principal chiral or sigma models, in the three-dimensional loop space formulation of Yang-Mills theory, and, so far, in a restricted class of four-dimensional local Yang-Mills gauge field theory, the self-dual set. The use of such infinite parameter invariance in finding exact non-perturbative solutions is discussed in terms of action-angle variables, the inverse scattering problem method and exact S-matrix calculations. Both statistical mechanical spin models and continuum quantum field theories are analyzed. Evidence and suggestions for the extension of the affine symmetry to the complete non-Abelian gauge theory are given.