How to deal with conformal and pure scale-invariant theories of gravity in d dimensions?

Conformally-invariant and pure, scale-invariant theories of gravity are particularly interesting in four or higher dimensions. Yet, in contrast to their four-dimensional counterparts, theories in higher dimensions are significantly more difficult to study. In these proceedings, following our recent work, we will formulate such theories in d dimensions, present an elegant way to handle them, and show that imposing invariance under scale or conformal transformations gives rise to entirely different properties when compared to their four-dimensional analogues.