The author describes a modified binomial method that provides a simple and unified framework for the valuation of various kinds of Asian options (American or European, arithmetic or geometric, fixed or floating strike, discrete or continuous sampling and dividends, and partial Asians). The greeks can also be calculated accurately and stably. The method is a refinement of that of Hull and White (1993), where at each node of a standard binomial tree one also considers a table of possible values of the average. To avoid the exponential explosion of the size of this table in the arithmetic average case, one considers a smaller set of representative values for the average, interpolates when necessary, and otherwise uses standard backward recursion to value the option. In this paper, an efficient implementation of this idea is presented. In particular, option values are insured to be smooth as a function of the number N of binomial time periods, so that Richardson extrapolation can be applied to eliminate 1/N (and sometimes higher-order) corrections, dramatically increasing the speed of the method. Detailed checks and illustrations are provided, showing that this approach can achieve any desired level of accuracy for convection- or diffusion-dominated regimes and for long or short maturities. It is typically much faster than standard PDE and Monte Carlo approaches.