New developments in the Renormalization Group

In the first part of the thesis, we review the basics of the Exact Renormalization Group. In the central part, we design a specific choice of renormalization scheme in the context of Functional Renormalization Group to achieve the nonperturbative analogous of the MS scheme's results. Then, we study the properties of a more general family of renormalization schemes, that includes the one we previously analyze, and appears to be useful to eliminate the spurious breaking of symmetries cause by the renormalization scheme. The final part of this thesis consists of a new implementation of the Functional Renormalization Group, based on the Effective Average Action, that allows all possible field redefinitions to simplify the flow equations. Such a simplification is practically useful in reducing the complexity of the computations and has theoretical implications in disentangling the unphysical information due to intrinsic redundancies of the mathematical descriptions of Nature. We show such improvements in the context of the three-dimensional Ising model and the Quantum Einstein Gravity without matter. In particular, using the derivative expansion in both cases we impose renormalization conditions that fix the value of the inessential couplings obtaining only the flow of the essential ones. With such a renormalization scheme, which is called Minimal Essential Scheme, the propagator does not develop additional poles when the truncation of the derivative expansion is increased. This way, we can select the desired universality classes, avoiding encountering instabilities and unitarity violations.