In a Yang-Mills-Higgs (YMH) Gauge Theory it is sometimes useful to choose a Higgs-field dependent basis of the Lie algebra. Among others, this turns the structure constants of the Yang-Mills curvatures into Higgs-field dependent structure functions. This process can be controlled in terms of a connection $\nabla$ that lives in a bundle over the Higgs target manifold. It measures non-trivial deformations of the YMH system: The theory is equivalent to an ordinary YMH gauge theory, iff $\nabla$ is flat, $R_\nabla = 0$; in this case, there exists a local field redefinition bringing the system into the standard YMH-form. We construct also theories with an inherently "curved target", $R_\nabla \neq 0$. It turns out that this leads to a gauge theory where the YM structural Lie algebra is replaced by a Lie algebroid over the Higgs manifold. We call this new class of consistent non-trivial deformations of ordinary YMH systems *Curved YMH Gauge Theories*.

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