  We study quantum effects in higher curvature extensions of general relativity
using the functional renormalisation group. New flow equations are derived for
general classes of models involving Ricci scalar, Ricci tensor, and Riemann
tensor interactions. Our method is applied to test the asymptotic safety
conjecture for quantum gravity with polynomial Riemann tensor interactions of
the form $\sim\int \sqrt{g} \,(R_{\mu\nu\sigma\tau}R^{\mu\nu\sigma\tau})^n$ and
$\sim\int \sqrt{g} \, R\cdot(R_{\mu\nu\sigma\tau}R^{\mu\nu\sigma\tau})^n$, and
functions thereof. Interacting fixed points, universal scaling dimensions, gaps
in eigenvalue spectra, quantum equations of motion, and de Sitter solutions are
identified by combining high order polynomial approximations, Pad\'e
resummations, and full numerical integration. Most notably, we discover that
quantum-induced shifts of scaling dimensions can lead to a four-dimensional
ultraviolet critical surface. Increasingly higher-dimensional interactions
remain irrelevant and show near-Gaussian scaling and signatures of weak
coupling. Moreover, a new equal weight condition is put forward to identify
stable eigenvectors to all orders in the expansion. Similarities and
differences with results from the Einstein-Hilbert approximation, $f(R)$
approximations, and $f(R,{\rm Ric}^2)$ models are highlighted and the relevance
of findings for quantum gravity and the asymptotic safety conjecture is
discussed.
