This article provides a review of risk and risk-value models that lead to decision making by explicitly trading off between risk and value. We begin with a discussion of a preference-dependent measure of risk that is compatible with traditional expected utility theory, and therefore lends itself to an examination of how individuals perceive risky gambles as well as how they choose among them. Further, this risk measure can be the basis for generalizing models of decision making based on the intuitively appealing notion of trading off between risk and value. Based on the two-attribute expected utility axioms, this risk-value framework has much more flexibility in modeling preferences than the traditional single-attribute expected utility theory. Specifically, we describe three useful classes of decision models: moments risk-value models, exponential risk-value models, and generalized disappointment models. These decision models unify two streams of research: one in developing preference models and the other in modeling risk judgments. They can also provide prescriptive guidance for those people who are willing to deviate from the traditional expected utility preference models and make their decisions based on risk-value trade-offs, as in financial decision making.