Numerical solutions obtained through the use of Computational Fluid Dynamics (CFD) are subject to discretization error, which is locally generated by truncation error. The discretization error is extremely difficult to properly estimate and this in turn leads to uncertainty over the quality of the numerical solutions obtained via CFD methods and the engineering functionals computed using these solutions. Adjoint error estimation techniques specifically seek to estimate the error in functionals, but are dependent upon accurate truncation error estimates. This work examines the application of new, single-grid, truncation error estimation procedures to the problem of adjoint error estimation for both the quasi-1D and 2D Euler equations. The new truncation error estimation techniques are based on local reconstructions of the computed solutions and comparisons are made for the quasi-1D study in order to determine the most appropriate solution variables to reconstruct as well as the most appropriate reconstruction method. In addition, comparisons are made between the single-grid truncation error estimates and methods based on uniformally refining or coarsening the underlying numerical mesh on which the computed solutions are obtained. A method based on an refined grid error estimate is shown to work well for a non-isentropic flow for the quasi-1D Euler equations, but all truncation error estimations methods ultimately result in over prediction of functional discretization error in the presence of a shock in 2D. Alternatives to adjoint methods, which can only estimate the error in a single functional for each adjoint solution obtained, are examined for the 2D Euler equations. The defection correction method and error transport equations are capable of locally improving the entire computed solution, allowing for error estimates in multiple functionals. It is found that all three functional discretization error estimates perform similarly for the same truncation error estimate, although the defect correction method is the most costly from a computational viewpoint. Comparisons are made between truncation error and adjoint weighted truncation error based adaptive indicators. For the quasi-1D Euler equations it is found that both methods are competitive, however the truncation error based method is cheaper as a separate adjoint solve is avoided. For the 2D Euler equations, the truncation error estimates on the adapted meshes suffer due to a lack of smooth grid transformations which are used in reconstructing the computed solutions. In order to complete this work, a new CFD code incorporating a variety of best practices from the field of Computer Science is developed as well as a new method of performing code verification using the method of manufactured solutions which is significantly easier to implement than traditional manufactured solution techniques.