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Quantization of spacetime by means of finite dimensional matrices is thebasic idea of fuzzy spaces. There remains an issue of quantizing time, however,the idea is simple and it provides an interesting interplay of various ideas inmathematics and physics. Shedding some light on such an interplay is the maintheme of this dissertation. The dissertation roughly separates into two parts.In the first part, we consider a mathematical aspect of fuzzy spaces, namely,their construction. We begin with a review of the construction of fuzzy complexprojective spaces CP^k (k=1,2,...) in relation to geometric quantization. Wethen present the construction of fuzzy S^4, utilizing the fact that CP^3 is anS^2 bundle over S^4. This method is also applicable to the case of fuzzy S^8.In the second part, we consider applications of fuzzy spaces to physics. Wefirst consider gravitational theories on fuzzy spaces, anticipating that theymay offer a novel way of regularizing spacetime dynamics. Particularly weobtain actions for gravity on fuzzy S^2 and fuzzy CP^2. We also discussapplication to M(atrix) theory. Introducing extra potentials, we show that thetheory has new brane solutions whose transverse directions are described byfuzzy S^4 and fuzzy CP^3. The extra potentials can be interpreted as fuzzyversions of differential forms or fluxes, which enable us to discusscompactification models of M(atrix) theory. Compactification down to fuzzy S^4is discussed and a physically interesting matrix model in four-dimensions isproposed.

Construction of Fuzzy Spaces and Their Applications to Matrix Models