  A basic problem of classical field theory, which has attracted growing
attention over the past decade, is to find and classify all nonlinear
deformations of linear abelian gauge theories.
  The first part of this paper summarizes and significantly elaborates a field-
theoretic deformation method developed in earlier work. As a key contribution
presented here, a universal geometrical structure common to a large class of
nonlinear gauge theory examples is uncovered. This structure is derived
geometrically from the deformed gauge symmetry and is characterized by a
covariant derivative operator plus a nonlinear field strength, related through
the curvature of the covariant derivative. The scope of these results
encompasses Yang-Mills theory, Freedman-Townsend theory, Einstein gravity
theory, in addition to their many interesting types of novel generalizations
that have been found in the past several years.
  The second part of the paper presents a new geometrical type of Yang-Mills
generalization in three dimensions motivated from considering torsion in the
context of nonlinear sigma models with Lie group targets (chiral theories). The
generalization is derived by a deformation analysis of linear abelian
Yang-Mills Chern-Simons gauge theory. Torsion is introduced geometrically
through a duality with chiral models obtained from the chiral field form of
self-dual 2+2 dimensional Yang-Mills theory under reduction to 2+1 dimensions.
Field-theoretic and geometric features of the resulting nonlinear gauge
theories with torsion are discussed.
