A viscous discrete adjoint approach to automatic aerodynamic shape optimization is developed, and the merits of the viscous discrete and continuous adjoint approaches are discussed. The viscous discrete and continuous adjoint gradients for inverse design and drag minimization cost functions are compared with finite-difference and complex-step gradients. The optimization of airfoils in two-dimensional flow for inverse design and drag minimization is illustrated. Both the discrete and continuous adjoint methods are used to formulate two new design problems. First, the timedependent optimal design problem is established, and both the time accurate discrete and continuous adjoint equations are derived. An application to the reduction of the time-averaged drag coefficient while maintaining time-averaged lift and thickness distribution of a pitching airfoil in transonic flow is demonstrated. Second, the remote inverse design problem is formulated. The optimization of a three-dimensional biconvex wing in supersonic flow verifies the feasibility to reduce the near field pressure peak. Coupled drag minimization and remote inverse design cases produce wings with a lower drag and a reduced near field peak pressure signature.