Grid-embedding techniques are presently applied to the solutions of the Euler equations on Cartesian grids for the NACA 0012 airfoil and the multielement SKF1.1 airfoil at transonic speeds. Based on comparisons with solutions on fine grids, it is shown that the present scheme, when used in conjunction with the Runge-Kutta time-stepping scheme as well as Cartesian grids, yields reduced memory requirements and a substantial reduction in computational time, without loss of accuracy.