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We present a concise description of the convex duality theory in this chapter. The goal is to lay a foundation for later application in various financial problems rather than to be comprehensive. We emphasize the role of the subdifferential of the value function of a convex programming problem. It is both the set of Lagrange multiplier and the set of solutions to the dual problem. These relationships provide much convenience in financial applications. We also discuss generalized convexity, conjugacy, and duality.