The authors review models of the option surface and reduced-form models for stochastic volatility in continuous time, under the risk-neutral measure. They introduce ``forward volatilities'' (in analogy with forward interest rates in the term structure theory), and prove that such objects are conditional expected values, under the risk-neutral measure, of the future spot volatility. The theory developed here is the analogue of Heath-Jarrow-Morton bond-pricing theory. Moreover, a link is established between forward volatilities and the so-called ``model-free'' volatility measures.