rlo simulation. First, we develop a mesh-based, biased-low estimator. By recursively averaging the low and high estimators at each stage, we obtain a significantly more accurate point estimator at each of the mesh points. Second, adapt the importance sampling ideas for simulation of European path-dependent options in Glasserman, Heidelberger, and Shahabuddin (1998a) to pricing of American options with a stochastic mesh. Third, we sketch generalizations of the mesh method and we discuss links with other techniques for valuing American options. Our empirical results show that the bias-reduced point estimates are much more accurate than the standard mesh-method point estimators. Importance sampling is found to increase accuracy for a smooth option-payoff functions, while variance increases are possible for non-smooth payoffs.