In this dissertation, the concerned study of the research dealts with various types of the boundary value problems of nonlinear fractional-order differential equations. Almost for all kinds of arbitrary order differential equations, this study emphasize on existence theory, various aspect of Ulam stability, lower and upper solutions, iterative schemes and analytical methods of perturbation. We have developed strong sufficient conditions for the uniqueness and existence of upper and lower solutions for the fractional differential equations, with the help of classical fixed point theorem and monotone iterative method. A strong attention has also been given to the numerical solution of fractional differential equations and frac tional partial differential equations. In this regard, some powerful and an efficient numerical techniques have been established for the approximate solutions of both linear and nonlinear fractional order differential equations. One of them, an established technique is based on monotone sequence and one of them from pertubation method is based on homotopy defini tion and an known deformation equation to obtain the solutions in the form of convergent series. Both the methods are interesting and efficient for solving nonlinear fractional-order differential equations. In pertubation method, we have developed a new method for FPDEs which is known as Optimal Homotopy asysmptotic method OHAM based on homotopy def inition and an known deformation equation. With the help of afore mentioned techniques and methods, we solved both linear and nonlinear ordinary as well as partial fractional order differential equations. We considered some fractional order differential equations for illustrative purposes and numerical approximations of their solutions are tabulated and plotted via MATLAB softwares. The numerical results obtained via aforesaid techniques, are compared with other standard techniques. Which shows, that how these techniques are more effective and reliable, than the standard ordinary differential equations solvers.