  We develop a holonomy reduction procedure for general Cartan geometries. We
show that, given a reduction of holonomy, the underlying manifold naturally
decomposes into a disjoint union of initial submanifolds. Each such submanifold
corresponds to an orbit of the holonomy group on the modelling homogeneous
space and carries a canonical induced Cartan geometry. The result can therefore
be understood as a `curved orbit decomposition'. The theory is then applied to
the study of several invariant overdetermined differential equations in
projective, conformal and CR-geometry. This makes use of an equivalent
description of solutions to these equations as parallel sections of a tractor
bundle. In projective geometry we study a third order differential equation
that governs the existence of a compatible Einstein metric. In CR-geometry we
discuss an invariant equation that governs the existence of a compatible
K\"{a}hler-Einstein metric.

Differential Geometry