Numerical optimization procedures have been employed for the design of airfoils in transonic flow based on the transonic small-disturbance (TSD) and Euler equations. A sequential approximation optimization technique was implemented for solving the design problem of lift maximization with wave drag and area constraints. A simple linear approximation was utilized for the approximation of the lift. Accurate approximations for sensitivity derivatives of the wave drag were obtained through the utilization of Nixon's coordinate straining approach. A modification of the Euler surface boundary conditions was implemented in order to efficiently compute design sensitivities without recreating the grid. Our design procedures experienced convergence problems for some TSD solutions, where the wave drag was found not to vary smoothly with the design parameters and consequently create local optimum problems. A procedure interchanging the role of the objective function and constraint, initially minimizing drag with a constraint on the lift was found to be effective in producing converged designs, usually in approximately 10 global iterations. This procedure was also shown to be robust and efficient for cases where the drag varied smoothly, such as with the Euler solutions. The direct lift maximization with move limits which were fixed absolute values rather than fractions of the design variables, was also found to be a reliable and efficient procedure for designs based upon the Euler equations. / Ph. D.