It is well known that quantisation of gravity within the conventional framework of quantum
field theory faces challenges. An intriguing novel prospect was put forward by S. Weinberg
in 1979 who suggested that the metric degrees of freedom of gravity could be quantised nonpertubatively provided that the theory becomes asymptotically safe (AS) at high energies. In this thesis we put forward a systematic search strategy to test the AS conjecture in four dimensional quantum gravity. Using modern renormalisation group (RG) methods and heat kernel techniques we derive the RG equations for gravitational actions that are formed from powers of the Ricci scalar and powers of the Ricci tensor. The non-linear fixed point equations are solved iteratively and exactly. We develop a sophisticated algorithm to express the fixed point iteratively, and to high order, in terms of its lower order couplings. We also evaluate universal scaling exponents and find that the relevancy of invariants at an asymptotically safe fixed point is governed by their classical mass dimension, providing structural support for the asymptotic safety conjecture. We also apply our findings to the physics of higher dimensional black holes. Most notably, we find that the seminal ultra-spinning Myers-Perry black holes cease to exist as soon as asymptotically safe RG corrections are taken into account. Further results and implications of our findings are discussed
