A method for adaptive refinement of a triangular mesh for solution of the steady and unsteady Euler equations is presented. An upwind, finite volume based on Roe"s flux difference splitting method is used to discretize the equations. By using advancing front method an initial regular Delaunay triangulation has been made. The ADAPTATION procedures involve mesh enrichment and mesh coarsening to either add points in high gradient regions of the flow or remove points where they are not needed, respectively, to produce solutions of high spatial accuracy at minimal cost. Steady transonic results are shown to be of high spatial accuracy, primarily in that the shock waves are very sharply captured. Unsteady results obtained for a moving shock wave in a two-dimensional domain show the precise enrichment and coarsening procedure. The results were obtained with large computational savings when compared to results for globally-enriched mesh with cells subdivided as many times as the finest cells of the adapted grid.