Motivated by the problems of optimal compensation of executives and of investment fund managers, we consider principal-agent problems in continuous time, when the principal\'s and the agent\'s risk-aversion are modeled by standard utility functions. The agent can control both the drift (the ``mean\") and the volatility (the ``variance\"") of the underlying stochastic process. The principal decides what type of contract/payoff to give to the agent. We use martingale/duality methods familiar from the theory of continuous-time optimal portfolio selection. Our results depend on whether the agent can control the drift independently of the volatility