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This thesis is devoted to the study of Quantum Field Theories (QFT) on fuzzyspaces. Fuzzy spaces are approximations to the algebra of functions of acontinuous space by a finite matrix algebra. In the limit of infinitely largematrices the formulation is exact. An attractive feature of this approach isthat it transparently shows how the geometrical properties of the continuousspace are preserved. In the study of the non-perturbative regime of QFT, fuzzyspaces provide a possible alternative to the lattice as a regularisationmethod. The thesis is divided into two parts. We perform Monte Carlosimulations of a $\lambda \phi^4$ theory on a 3-dimensional Euclidean space. Weidentify the phase diagram of this model. In addition to the usual disorderedand uniform ordered phases we find a third phase of non-uniform ordering. Thisindicates the existence of the phenomenon called UV-IR mixing in the strongcoupling regime. Second we present a geometrical analysis of the scalar fieldtheory on a 4-dimensional fuzzy sphere, S4_F. Nevertheless a fuzzy version ofS4 cannot be achieved by quantisation of the classical space. The problem iscircumvented by defining a scalar theory on a larger space, CP3 which is6-dimensional. It includes degrees of freedom related to S^4 plus others beyondS4. Those extra degrees of freedom are dynamically suppressed through aprobabilistic method. The analysis of the geometrical structures allows us tointerpret this procedure as a Kaluza-Klein reduction of CP3 to S4.

Fuzzy Scalar Field Theories: Numerical and Analytical Investigations