Implementations of numerical simulations for solving systems of partial differential equations are often not verified and are falsely assumed to work correctly. As a result, the implementations are prone to coding errors that could degrade the accuracy of the solution. In order to ensure that a code is written correctly, rigorous verification of all parts of the code is necessary. Code verification is the task of ascertaining whether a numerical algorithm is solving the governing equations of the problem correctly. If an exact solution existed for the governing equations then verification would be easier but these solutions are rare because of the non-linearity of common Computational Fluid Dynamics (CFD) problems. In the absence of exact solutions, grid refinement studies are the most commonly used methods to verify codes using simulations on a sequence of grids but even these studies have limitations. The Method of Manufactured Solutions (MMS) is a novel and a recently developed technique that verifies the observed order-ofuracy of the implementation of a numerical algorithm. The method is more general and overcomes many of the limitations of the method of exact solutions and grid refinement studies. The central idea is to modify the governing equations and the boundary conditions by adding forcing functions or source terms in order to drive the discrete solution to a prescribed or ``manufactured' solution chosen a priori. A grid convergence study is performed subsequently to determine the observed orders. Two methods of accuracy assessment are presented here - solution accuracy analysis and residual error analysis. The method based on the error in the spatial residual is computationally less expensive and proved to be a valuable debugging tool. In the present work, the Method of Manufactured Solutions (MMS) is implemented on a compressible flow solver that solves the two-dimensional Euler equations on structured grids and an incompressible code that solves the two-dimensional Navier-Stokes equations on unstructured meshes. Exponential functions are used to ``manufacture' steady solutions to the governing equations. Solution and residual error analyses are presented. The influence of grid non-uniformity on the numerical accuracy is studied.