An aerodynamic design method for turbomachine blades using robust time-marching algorithms for the numerical solutions of the Euler equations is proposed. In this inverse method, the blade loading and thickness distributions are prescribed, and the corresponding blade geometry is sought after. A four-stage Runge-Kutta time-stepping scheme is used to march the finite-volume formulation of the unsteady Euler equations to steady-state. The inverse problem is formulated using a pressure-loading boundary condition across the blade surfaces, and modification of the blade geometry is achieved using the flow-tangency conditions along the blade surfaces. The method is demonstrated for the design of two-dimensional cascaded blades ranging from the subsonic to the supersonic flow regimes.