Quantum field theory is the underlying framework of most of our progress in modern particle physics and has been succesfully applied also to statistical mechanics and cosmology. A basic concept of quantum field theory is the renormalization group which describes how physics changes according to the energy at which we probe the system. The functional renormalization group (fRG) for the effective average action (EAA) describes the Wilsonian integration of high momentum modes without expanding in any small parameter. As such this is a non-perturbative framework and can be used to obtain non-perturbative insights, even though some other approximations are necessary. However we are not assured that quantum field theory is the correct framework to describe physics up to arbitrary high energies. This may happen if the theory approaches an ultra violet fixed point so that all physical quantities remain finite. In this case predictivity requires a finite number of relevant directions in such a way that only a finite number of parameters needs to be fixed by the experiments.
In this thesis we consider the fRG to address several problems. In chapter 1 we briefly review the fRG for the EAA deriving its flow equation and describing how theories with local symmetries can be handled and possible strategies of computation. In chapter 2 we describe how this framework can be used to investigate whether a quantum theory of gravity can be consistently built within the framework of standard quantum field theory. In particular we consider a new approximation of the flow equation for the EAA where the difference between the anomalous dimension of the fluctuating metric and the Newton’s constant is taken into account. In chapter 3 we show that Weyl invariance can be maintained along the flow if a dilaton is present and if a judicious choice of the cutoff is made. This seems to contradict the standard lore according to which the renormalization group breaks Weyl invariance introducing a mass scale which is the origin of the so called trace anomaly. We analyze this in detail and show that standard results can be reobtained in a specific choice of gauge. Finally in chapter 4 we discuss a global feature of the renormalization group in two dimensions: the c-theorem. This is a global feature of the RG since it regards the whole RG trajectory from the UV to the IR. In particular we derive an exact equation for the c-function and, with some approximations, compute it explicitly in some examples. This also leads to some insights about a generic form of a truncation for the EAA. Some background material and technical details are confined to several appendices at the end of the thesis.