Gravity beyond Einstein? Yes, but in which direction?

We present qualitative arguments in favour of an extension of the theory of the gravitational interaction beyond that resulting from the Hilbert-Einstein action. To this end, we consider a locally conformal invariant theory of gravity, discussed some 30 years ago by Mannheim and Kazanas. We discuss its exact solution of the static, spherically symmetric configurations and, based on these, we revisit some of the outstanding problems associated with gravity, high energy interactions and sketch potential resolutions within the conformal gravity framework. This article is part of the theme issue 'The future of mathematical cosmology, Volume 2'.