The availability of high frequency data for financial instruments has opened the possibility of accurately determining volatility in small time periods, such as one day. Recent work on such estimation indicates that it is necessary to analyze the data with a hidden semimartingale model, typically by the addition of measurement error. We review the emerging theory on this subject, including two- and multiscale sampling. We also consider broader error schemes, through Markov kernels and such phenomena as rounding due to discreteness of prices. Finally, we discuss the possibility of adapting likelihood theory to inference problems of this type.