  Finite dimensional truncations and the single field approximation have thus
far played dominant roles in investigations of asymptotic safety for quantum
gravity. This thesis is devoted to exploring asymptotic safety in infinite
dimensional, or functional, truncations of the effective action as well as the
effects that can be caused by the single field approximation in this context.
It begins with a comprehensive analysis of the three existing flow equations of
the single field f(R) truncation by determining their spaces of global fixed
point solutions and, where applicable, of corresponding eigenoperator
solutions. As a second result, it is then shown that one incarnation of the
single field f(R) approximation actually breaks down in the sense that there is
no physical content left to explore. In order to clarify whether such drastic
findings can be caused by the approximations used in setting up the
renormalisation group flow, we identify the single field approximation as a
prime candidate and show in the more familiar context of scalar field theory
that it can indeed lead to many types of non-physical results. As a way to
avoid such non-physical behaviour we highlight the importance of the previously
known split Ward identity and exemplify its usefulness by fully restoring the
correct physical picture in scalar field theory. Taking this result as evidence
that the split Ward identity may lead to well behaved functional truncations
also in gravity, we derive the flow equations of conformal gravity in a
bi-field truncation of the effective action that goes beyond the local
potential approximation in the fluctuation field. It is found that the split
Ward identity leads to a simplified set of renormalisation group equations for
the conformal factor that, while differing at crucial points, bear close
resemblance to flow equations obtained in scalar field theory.
