In this thesis we study examples of critical behavior of quantum systems coupled to a background metric. The analysis is performed with the double purpose of extracting information about matter degrees of freedom and gravitational degrees of freedom. Different formulations of the renormalization group are reviewed and applied to extract the critical properties of the systems under examination. In particular we focus on two classes of systems. First we analyze the behavior of chiral fermionic matter on negatively curved space which typically exhibit an enhanced tendency towards chiral symmetry breaking due to the background curvature. In this thesis we argue that, due to the high energies involved near the Planck regime, this phenomenon could result in the generation of heavy fermionic particles; inconsistent with the current observations of particle physics. We perform a scale dependent analysis to identify a set of parameters which rules out such a scenario and can lead to a criterion to for the falsification of models of quantum gravity. We claim that any theory of quantum gravity that admits a formulation in terms of a local field theory of the metric should respect our constraint or provide an alternative mechanism to solve the problem. The second class of systems we study is a tower of infinite scalar models with multicritical properties. Here we analyze their renormalization ow at leading order employing a covariant version of the epsilon expansion in dimensional regularization. We then investigate their analytical extension below the upper critical dimension paying particular attention to the fixed point value of the non minimal coupling to the background, which turns out to be the conformal value. Thanks to an improved parametrization of the action we can test the validity of our analysis against known results in the context of two dimensional quantum gravity couple to a large number of scalar fields and of statistical field theory in curved space
