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In this short course we study a problem of maximizing the expected robust utility from terminal wealth. Duality methods provide a powerful approach to this problem under very general assumptions on a model. Our aim is to present a collection of mathematical ideas and methods used to realize this approach in full generality. The topics to be discussed are: - minimax theorems reducing the problem to the case of standard utility; - description of a dual problem, existence and uniqueness of solutions, dual characterization of the value function of the primal problem; - conditions for the existence of a solution to the primal problem; - connections to the arbitrage theory and super-replication prices. For the most part of the course, we deal with an abstract (static or dynamic) setting of the problem which needs no prerequisites, though some familiarity with basic notions of functional and convex analysis (Banach spaces, weak topologies, Fenchel transform) may be helpful. From stochastic calculus, only the notion of a supermartingale is necessary for understanding.