Summary. The problem of evaluating European options under standard geometric Brownian motion is quite well developed even in the case of multifactor exotic options. It was realized early in the literature on option pricing that the standard geometric Brownian motion assumption was not adequate to capture various market features. As a result the dynamics for the underlying asset have been extended to incorporate stochastic volatility and jump components. The problem of evaluating American options (that is those that can be exercised at any time prior to maturity) is essentially a free boundary value problem. The literature on this problem goes back to the classical paper of McKean (1967). For the case of standard options, even when the underlying asset follows stochastic volatility and jump diffusion dynamics, this problem may be regarded as solved. In the case of exotic type American options under stochastic volatility and jump diffusion dynamics, there are still many challenges. It is true that the problem may be resolved by use of Monte Carlo simulation but it is useful to have alternative numerical scheme for low dimensional problems that may be used as a benchmark for the Monte Carlo methods. This talk will survey recent results into the representation and methods for numerical evaluation of two dimensional exotic American options under stochastic volatility and jump diffusion dynamics such as spread options, swing options and max options. The talk will cover both representation of the solution using transform methods and probabilistic methods as wells discuss numerical schemes such as the method of lines, operator splitting methods and spectral expansions. The level of presentation will make the material accessible to non finance specialists.