Many models in physics are inherently discrete at the very smallest scale (typically the size of an atom) and there is inherent randomness at this discrete level. At the macroscopic level the randomness is typically not seen. But under certain conditions this randomness manifests itself at the macroscopic scale. This is known as a critical phenomena. For example, the spins in a magnetic material have randomness in their orientation. At a certain temperature these spins can align with each other to produce macroscopic magnetic domains that play a crucial role in technological applications such as hard drives. The randomness seen at the macroscopic scale often does not depend on the details of the microscopic randomness. In physics this is called universality and has developed into a key idea in the understanding of critical phenomena. The renormalization group is set of methods from physics that has become the basis for the modern understanding of both critical phenomena and of quantum field theory - the theory of elementary particles. Despite the tremendous success of the renormalization group in physics, we do not have a deep mathematical understanding of this set of ideas. This research will further the mathematical development and understanding of this set of ideas and methods and use them to understand the mathematics of critical phenomena in models such as self-avoiding random walks and Ising models of magnetic spins.Most of the research will be devoted to random walk models and Ising-type models. The smart kinetic walk (also known as a limiting case of the Laplacian-b random walk) is a dynamic model for self-avoiding walks. On the hexagonal lattice it is closely related to percolation. Percolation methods have been used to prove its scaling limit is the Schramm-Loewner evolution with parameter value 6. The research will study this scaling limit on other lattices and for generalizations of the transition probabilities. The goal is to understand the universality of this scaling limit. For the ordinary random walk the scaling limit of the exit distribution for a domain is harmonic measure. Another goal of the research is to understand the first order correction for this convergence for both the ordinary random walk and the smart kinetic walk. The research will also study real space renormalization groups for Ising type models, in particular exact renormalization group transformations in one and two dimensions. These transformation have the potential advantage that the map would act on a finite dimensional space. The goal is to prove that the map can be rigorously defined and then take advantage of the finite dimensional nature to prove existence of a fixed point and all the exciting consequences that follow from this existence. Finally the research will study Schramm-Loewner evolution as a renormalization group fixed point. This stochastic process is known to be the scaling limit of many critical models. The goal here is to define a renormalization group map that has this process as a fixed point and then use this map to understand models that are near criticality.