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In this thesis we investigate various fundamental aspects of asymptoticallysafe quantum gravity, in particular the compatibility of Asymptotic Safety withthe requirements for background independence and unitarity. The first partcontains a detailed analysis of the space of metrics. We unveil a novel,specifically designed connection on this space and use the correspondinggeodesics to connect all metrics having the same signature. In the second partwe study the consequences of different metric parametrizations, i.e. ofdifferent connections on the space of metrics, for the Asymptotic Safetyscenario by exploring the respective properties of the decisive renormalizationgroup fixed points. Based on a bimetric investigation we show that AsymptoticSafety can be reconciled with background independence. The third part of thiswork focuses on the 2D limit of quantum gravity. To this end, we show that the2D limit of the Einstein--Hilbert action at a nontrivial fixed point becomesPolyakov's induced gravity action which describes a conformal field theory. Inthis way, we can prove that the fixed point theory is unitary. The last partconcerns the reconstruction of the functional integral out of a given effectivetheory, where we reconstruct the bare action for an Einstein--Hilbert-typeeffective average action and for a Liouville action.

Field space parametrization in quantum gravity and the identification of a unitary conformal field theory at the heart of 2D Asymptotic Safety