The Banach contraction principle states that a contraction on a complete metric space has a unique fixed point and its proof hinges on "Picard iterations". This principle is applicable to a variety of subjects such as integral equations, partial differential equations and engineering of image processing. This principle fails for nonexpansive mappings on a Banach space. Mann [Proc. Amer. Math. Soc. 4(1953), 506-510] introduced an iterative scheme to approximate fixed points of a nonexpansive mapping on a Banach space. Mann scheme is inadequate for the approximation of fixed points of pseudocontractive mappings even on a Hilbert space. Consequently, Ishikawa [Proc. Amer. Math. Soc., 44 (1974), 147-150] upgraded Mann iterative scheme which is extensively used to approximate common fixed points of nonlinear mappings including nonexpansive mappings. The purpose of this dissertation is two-fold: (i) To prove the existence of fixed point for the classes of nonlinear mappings, namely generalized nonexpansive mappings and generalized quasi- contractive mappings in the setting of uniformly convex metric spaces. (ii) To establish approximate fixed point property for the classes of nonlinear mappings, namely generalized nonexpansive mappings, asymptotically nonexpansive mappings and generalized quasi-contractive mappings in the setting of (uniformly / strictly) convex metric spaces and cone metric type spaces. The approximation of fixed points is obtained by using appropriate iterative schemes; for example, the averaged iteration scheme, Ishikawa iteration scheme, multi-step iterative scheme, three-step explicit iterative scheme and Jungck three-step implicit iterative scheme. The strong convergence analysis of different iterative schemes contribute significantly in metric fixed point theory of nonlinear mappings. Most of the results presented here are new in the setting of a metric space and are: (a) established with the limited set of conditions on the control parameters (b) supported by real world applications, such as the existence of a solution for the first order periodic boundary value problem and the existence of a solution for an implicit integral equation.