  We study the renormalization group flow of $\mathbb{Z}_2$-invariant
supersymmetric and non-supersymmetric scalar models in the local potential
approximation using functional renormalization group methods. We focus our
attention to the fixed points of the renormalization group flow of these
models, which emerge as scaling solutions. In two dimensions these solutions
are interpreted as the minimal (supersymmetric) models of conformal field
theory, while in three dimension they are manifestations of the Wilson-Fisher
universality class and its supersymmetric counterpart. We also study the
analytically continued flow in fractal dimensions between 2 and 4 and determine
the critical dimensions for which irrelevant operators become relevant and
change the universality class of the scaling solution. We also include novel
analytic and numerical investigations of the properties that determine the
occurrence of the scaling solutions within the method. For each solution we
offer new techniques to compute the spectrum of the deformations and obtain the
corresponding critical exponents.
