Quantum gravity and the renormalisation group: from the UV to the IR

General relativity is the successful classical theory describing gravitational interactions from cosmological scales down to the sub-millimetre scale. It has remained an open challenge to combine the principles of general relativity with those of the quantum world. A promising avenue has been put forward by Steven Weinberg, known as the asymptotic safety conjecture for gravity. It stipulates that a quantum field theory of gravity may very well exist as a fundamental and predictive theory up to highest energies. The central ingredient of this scenario is the existence of an interacting ultraviolet fixed point under the renormalisation group running of gravitational couplings. In this thesis, we study several aspects of asymptotic safety for gravity. Firstly, we offer a detailed qualitative and quantitative analysis of modern renormalisation group equations for Einstein-Hilbert gravity by contrasting different implementations of a Wilsonian momentum cutoff in combination with either heat kernel techniques or spectral sums. Secondly, we analyse in some depth the scale-dependence of gravitational couplings in the low-energy regime of Einstein-Hilbert gravity, where indications for the existence of an interacting infrared fixed point are found. Finally, we extend our analysis of renormalisation group trajectories to f(R)-type theories of gravity, and investigate how an interacting UV fixed point is connected with the classical low-energy regime. Implications of our findings are discussed.