  This work has its origins in an attempt to describe systematically the
integrable geometries and gauge theories in dimensions one to four related to
twistor theory. In each such dimension, there is a nondegenerate integrable
geometric structure, governed by a nonlinear integrable differential equation,
and each solution of this equation determines a background geometry on which,
for any Lie group $G$, an integrable gauge theory is defined. In four
dimensions, the geometry is selfdual conformal geometry and the gauge theory is
selfdual Yang-Mills theory, while the lower-dimensional structures are
nondegenerate (i.e., non-null) reductions of this. Any solution of the gauge
theory on a $k$-dimensional geometry, such that the gauge group $H$ acts
transitively on an $\ell$-manifold, determines a $(k+\ell)$-dimensional
geometry ($k+\ell\leqslant4$) fibering over the $k$-dimensional geometry with
$H$ as a structure group. In the case of an $\ell$-dimensional group $H$ acting
on itself by the regular representation, all $(k+\ell)$-dimensional geometries
with symmetry group $H$ are locally obtained in this way. This framework
unifies and extends known results about dimensional reductions of selfdual
conformal geometry and the selfdual Yang-Mills equation, and provides a rich
supply of constructive methods. In one dimension, generalized Nahm equations
provide a uniform description of four pole isomonodromic deformation problems,
and may be related to the ${\rm SU}(\infty)$ Toda and dKP equations via a
hodograph transformation. In two dimensions, the ${\rm Diff}(S^1)$ Hitchin
equation is shown to be equivalent to the hyperCR Einstein-Weyl equation, while
the ${\rm SDiff}(\Sigma^2)$ Hitchin equation leads to a Euclidean analogue of
Plebanski's heavenly equations.
