  This thesis is devoted to the study of Quantum Field Theories (QFT) on fuzzy
spaces. Fuzzy spaces are approximations to the algebra of functions of a
continuous space by a finite matrix algebra. In the limit of infinitely large
matrices the formulation is exact. An attractive feature of this approach is
that it transparently shows how the geometrical properties of the continuous
space are preserved. In the study of the non-perturbative regime of QFT, fuzzy
spaces provide a possible alternative to the lattice as a regularisation
method. The thesis is divided into two parts. We perform Monte Carlo
simulations of a $\lambda \phi^4$ theory on a 3-dimensional Euclidean space. We
identify the phase diagram of this model. In addition to the usual disordered
and uniform ordered phases we find a third phase of non-uniform ordering. This
indicates the existence of the phenomenon called UV-IR mixing in the strong
coupling regime. Second we present a geometrical analysis of the scalar field
theory on a 4-dimensional fuzzy sphere, S4_F. Nevertheless a fuzzy version of
S4 cannot be achieved by quantisation of the classical space. The problem is
circumvented by defining a scalar theory on a larger space, CP3 which is
6-dimensional. It includes degrees of freedom related to S^4 plus others beyond
S4. Those extra degrees of freedom are dynamically suppressed through a
probabilistic method. The analysis of the geometrical structures allows us to
interpret this procedure as a Kaluza-Klein reduction of CP3 to S4.
