This paper investigates the Multifractal Model of Asset Returns, a class of continuous-time processes that incorporate the thick tails and volatility persistence exhibited by many financial time series. The simplest version of the model compounds a Brownian Motion with a multifractal time-deformation process. Prices follow a semi-martingale, which precludes arbitrage in a standard two-asset economy. Volatility has long memory, and the highest finite moments of returns can take any value greater than two. The local variability of the process is highly heterogeneous, and is usefully characterized by the local Hölder exponent at every instant. In contrast with earlier processes, this exponent takes a continuum of values in any time interval. The model also predicts that the moments of returns vary as a power law of the time horizon. We confirm this property for Deutsche Mark/U.S. Dollar exchange rates and several equity series. We then develop an estimator, and infer a parsimo...