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This book is related to a topic in functional analysis called "fixed point theory". It is an outcome of the study of fixed point theorems in different spaces. In Chapter I, we give the background of fixed point theory. In the Chapter II, we obtain common fixed point theorems for weakly compatible maps on symmetric spaces.In the second section of this chapter we obtain common fixed point theorems of occasionally weakly compatible maps on (X, d) under relaxation on d. In this chapter we also study the application of fixed point theory in dynamic programming. In the Chapter III, we study common fixed theorems for multivalued maps. In the Chapter IV, common fixed point theorems for a pair of occasionally weakly compatible maps in probabilistic semi - metric space have been obtained. In the second section we obtain some fixed point theorems for JH-operators and occasionally weakly g-biased maps on a set X together with the function F. In Chapter V, we give the epsilon delta contractive definition in the setting of a fuzzy metric space.