  In this paper we introduce a set of equations on a principal bundle over a
compact complex manifold coupling a connection on the principal bundle, a
section of an associated bundle with K\"ahler fibre, and a K\"ahler structure
on the base. These equations are a generalization of the K\"ahler-Yang-Mills
equations introduced by the authors. They also generalize the constant scalar
curvature for a K\"ahler metric studied by Donaldson and others, as well as the
Yang-Mills-Higgs equations studied by Mundet i Riera. We provide a moment map
interpretation of the equations, construct some first examples, and study
obstructions to the existence of solutions.
