We develop lower and upper bounds on the prices of American call and put options written on a dividend-paying asset. We provide two option price approximations one based on the lower bound (termed LBA) and one based on both bounds (termed LUBA). The LUBA approximation has an average accuracy comparable to a l,000-step binomial tree. We introduce a modification of the binomial method (termed BBSR) that is very simple to implement and performs remarkably well. We also conduct a careful large-scale evaluation of many recent methods for computing American option prices.