Quantization of General Relativity is ill-fated for many reasons, and we showcase a particularly severe problem of the wrong-sign kinetic term of the conformal mode in the Euclidean path integral. On the other hand, the quantization of matter fields in a classical gravitational background is relatively well understood; in this chapter, we focus on the d=2 Weyl anomaly and the gravitational effective action that can be derived from the former, which is possible thanks to how the d=2 metric can always be recast into a conformally flat form. After a detailed derivation of the resulting Liouville action, based on the heat kernel expansion, we find an application of the Weyl anomaly in the context of black holes. The energy-momentum expectation values are entirely fixed by the usual conservation law when combined with the Weyl anomaly, and this leads to the late-time Hawking radiation under the initial condition of an empty future event horizon. We then move on to how the Euclidean action of black holes comes with a precise interpretation as a thermodynamic free energy with definite temperature and entropy, and close with a discussion of how various ensembles are realized in this setting.