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Wide\u2010ranging applications to financial data have discovered important stylized facts and illustrated both the strengths and weaknesses of the models. There are now many surveys of this literature. This paper looks forward to identify promising areas of new research. The paper lists five new frontiers. It briefly discusses three\u2014high\u2010frequency volatility models, large\u2010scale multivariate ARCH models, and derivatives pricing models. Two further frontiers are examined in more detail\u2014application of ARCH models to the broad class of non\u2010negative processes, and use of Least Squares Monte Carlo to examine non\u2010linear properties of any model that can be simulated. Using this methodology, the paper analyses more general types of ARCH models, stochastic volatility models, long\u2010memory models and breaking volatility models. The volatility of volatility is defined, estimated and compared with option\u2010implied volatilities. 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