In this chapter, we first develop the Karush-Kuhn-Tucker (KKT) theory for nonsmooth convex minimization problems. This is achieved by leveraging the characterization of the subdifferential of convex functions through their directional derivatives, as well as the formula for the subdifferential of maximum functions. We then present several applications of the KKT theory, including the derivation of the well-known Hoffman error bound. Next, we explore Lagrangian duality theory and its various applications. In particular, we investigate the dual proximal gradient method and the augmented Lagrangian method.