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Self-organizing patterns arise in a variety of ways in nature, the complex patterning observed on animal coats is such an example. It is already known that the mechanisms responsible for pattern formation starts at the developmental stage of an embryo. However, the actual process determining cell fate has been, and still is, unknown. The mathematical interest for pattern formation emerged from the theories formulated by the mathematician and computer scientist Alan Turing in 1952. He attempted to explain the mechanisms behind morphogenesis and how the process of spatial cell differentiation from homogeneous cells lead to organisms with different complexities and shapes. Turing formulated a mathematical theory and proposed a reaction-diffusion system where morphogens, a postulated chemically active substance, moderated the whole mechanism. He concluded that this process was stable as long as diffusion was neglected; otherwise this would lead to a diffusion-driven instability, which is the fundamental part of pattern formation. The mathematical theory describing this process consists of solving partial differential equations and Turing considered deterministic reaction-diffusion systems.   This thesis will start with introducing the reader to the problem and then gradually build up the mathematical theory needed to get an understanding of the stochastic reaction-diffusion systems that is the focus of the thesis. This study will to a large extent simulate stochastic systems using numerical computations and in order to be computationally feasible a compartment-based model will be used. Noise is an inherent part of such systems, so the study will also discuss the effects of noise and morphogen kinetics on different geometries with boundaries of different complexities from one-dimensional cases up to three-dimensions.