In this thesis I introduce a new methodology for pricing American options when the underlying model of the asset price allows for stochastic volatility and/or it has a multi-factor structure. Our approach is based on a decomposition of an American option price into its European options counterpart price and the early exercise premium, paid by the option holder in order to keep the right of exercising the option at any time-point before its expiration date. Based on closed form solutions of the joint characteristic function of the state variables driving the underlying model, the thesis provides analytic, integral solutions of the early exercise premium (and hence of the American option price) which enable us to build up fast and accurate numerical approximation procedures for calculating options prices. The analytic solutions that I derive express the optimal early exercise boundary in terms of prices of Arrow-Debreu type of securities reflecting the values of the options additional payoffs if they are exercised earlier, or not. Numerical results reported in the thesis show that our approach can price American options on stocks, bonds and interest rates derivatives efficiently and very fast, compared with existing methods. The efficiency gains of our method stem from the fact that it involves only one step of approximation, as the European prices embodied in the American option prices can be calculated analytically. The gains of computational speed come from the fact that our method can reduce the integral dimensions of the early exercise premium considerably.