In this talk, I will give a survey of some extremal problems in various analytic function spaces. I will start by discussing the duality relationships that were first applied to such problems by S. Ya. Khavinson in the east and simultaneously by Rogosinski and Shapiro in the west in the late 1940s and early 1950s. I will then show how one can use this duality to establish the generic form of extremal solutions in both Hardy and Bergman spaces. I will discuss some explicit solutions in these spaces as well as in Fisher spaces and will show why the search for explicit solutions naturally leads to non-linear problems, for non-vanishing functions. Many open problems still exist in this area, the most famous of which is the Krzyz conjecture. Finally, I will discuss some current joint ongoing work with D. Khavinson on a specific problem in Bergman spaces.