The author gives a survey on the valuation problem for American options based on a risky asset which is modelled as a geometric Brownian motion. The fact that American options --- by definition --- can be exercised at any time up to a fixed maturity \(T\) makes the pricing problem more difficult than that for European options. The emphasis of the survey is to exhibit the relations of the problem to optimal stopping, free boundary problems and variational inequalities. The reader should have some basic knowledge on stochastic analysis (linear SDE, Girsanov theorem, Itô's formula, local time). The paper including the appendix contains many (62) references to the existing literature e.g. on numerical methods.