To study intertemporal decisions under risk, we develop a new recursive model of non-expected-utility preferences. The main axiom of our analysis is called mixture aversion, as it captures a dislike of probabilistic mixtures of lotteries. Our representation for mixture-averse preferences can be interpreted as if an individual optimally selects her risk attitude from some feasible set. We describe some useful parametric examples of our representation and provide comparative statics that tightly link decreases in risk aversion to larger sets of feasible risk attitudes. We then present several applications of the model. In an insurance problem, mixture-averse preferences can produce a marginal willingness to pay for insurance coverage that increases in the level of existing coverage. In investment decisions, our model can generate endogenous heterogeneity in equilibrium stock market participation, even when consumers have identical preferences. Finally, we demonstrate that our model can address the Rabin paradox even in the presence of reasonable levels of background risk.