This paper investigates the multifractal model of asset returns (MMAR), 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 MMAR compounds a Brownian motion with a multifractal time-deformation. 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 2. The local variability of a sample path is highly heterogeneous and is usefully characterized by the local Holder exponent at every instant. In contrast with earlier processes, this exponent takes a continuum of values in any time interval. The MMAR 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 develop an estimation procedure and infer a parsimonious generating mechanism for the e...