In this thesis, we use the Optimal Homotopy Asymptotic Method (OHAM), Differential Tranform Method (DTM), Multistage Optimal Homotopy Asymptotic Method (MOHAM), Adomian decomposition method (ADM) and Homotopy analysis method (HAM) for the solution of nonlinear initial and boundary value problems for ordinary and partial differential equations. The obtained results are compared with the results obtained from these methods and numerical results. These method do not required the linearization and discritization techniques like numerical methods andestablished better accuracy at low order approximation and its accuracy increases with increase in approximation orders; it uses a flexible auxiliary function that control the convergence of the solution and convergence region can easily be adjusted. Moreover, the procedure of these method are simple, well defined, and can easily be used the recursive relations explicitly. Except from the application of these methods, we have formulated some fluid problems and its solutions have been done. The physical interpretations of the parameters are discussed in detail. The conclusion of each model is given at the end of each chapter. The convergence regions for each problem have been mentioned graphically and the convergence is discussed. Also we have developed a new scheme for namely MOHAM which gives better accuracy not even for small domain problems but works very well for large domain boundary value problems. The implementation of this scheme is almost simple as the optimal homotopy asymptotic method.To shows its effectiveness we have used it to different bench mark problems from literature and compare the results with those obtained by other method. Fordeterminationofoptimalvaluesofconstantswe usemethodofleastsquareand collocation method for OHAM and MOHAM. For HAM h curves determine the convergence region and rate of convergence. We use Mathematica 9 for symbolic computation. Most of theworkpresentedinchapters2, 3,4,and5ofthisthesishasbeen publishedin different well reputed international journals and the remaining are submitted for possible publications. The details of published/accepted/submitted are included in the list of publications.