this book provides a concise introduction to convex duality in financial mathematics. convex duality plays an essential role in dealing with financial problems and involves maximizing concave utility functions and minimizing convex risk measures. recently, convex and generalized convex dualities have shown to be crucial in the process of the dynamic hedging of contingent claims. common underlying principles and connections between different perspectives are developed; results are illustrated through graphs and explained heuristically. this book can be used as a reference and is aimed toward graduate students, researchers and practitioners in mathematics, finance, economics, and optimization. topics include: markowitz portfolio theory, growth portfolio theory, fundamental theorem of asset pricing emphasizing the duality between utility optimization and pricing by martingale measures, risk measures and its dual representation, hedging and super-hedging and itsrelationship with linear programming duality and the duality relationship in dynamic hedging of contingent claims