A basic theoretical and practical problem in Mathematical Finance is how to hedge derivative securities. For instance, a financial institution may have sold a call option and thus have the obligation of paying out an amount of money depending on a stock price at a given time in the future. It wants to hedge this risk by dynamically trading in the stock. If the stock price follows the celebrated Black-Scholes model, then there is a trading strategy in the stock that exactly replicates the value of the call option---thus all risk is eliminated. In reality, the Black-Scholes model is a sometimes inadequate approximation of the behavior of stock prices. One popular modification is to allow for stochastic volatility. In such a model, it is no longer possible to perfectly hedge a call option. It is still possible to be "on the safe side", using superhedging strategies, but these are often too expensive to be practical. Another approach, which is the one that we take, is partial hedging, where the institution is prepared to take some risk, but where the loss is, in some sense, minimized. This is related to state-dependent utility maximization problems in classical economics. We derive the dual problem from the Legendre transform of the associated Hamilton-Jacobi-Bellman equation and interpret the optimal strategy as the perfect hedging strategy for a modified claim. Under the assumption that volatility is fast mean-reverting we derive, using a singular perturbation analysis, approximate value functions and strategies that are easy to implement and study. The analysis identifies the usual mean historical volatility and the harmonically-averaged long-run volatility as important statistics for such optimization problems without further specification of a stochastic volatility model. The approximation can be improved by specifying a model and calibrated for the leverage effect from the implied volatility skew. We study the effectiveness of these strategies using simulated stock paths. Joint work with Mattias Jonsson (University of Michigan).