  Constructs from conformal geometry are important in low dimensional gravity
models, while in higher dimensions the higher curvature interactions of
Lovelock gravity are similarly prominent. Considering conformal invariance in
the context of Lovelock gravity leads to natural, higher-curvature
generalizations of the Weyl, Schouten, Cotton and Bach tensors, with properties
that straightforwardly extend those of their familiar counterparts. As a first
application, we introduce a new set of conformally invariant gravity theories
in D=4k dimensions, based on the squares of the higher curvature Weyl tensors.
