We review recent advances in the field of decision making under uncertainty or ambiguity. We start with a presentation of the general approach to a decision problem under uncertainty, as well as the 'standard' Bayesian treatment and issues with this treatment. We present more general approaches (Choquet expected utility, maximin expected utility, smooth ambiguity and so forth) that have been developed in the literature under the name of models of ambiguity sensitive preferences. We draw a distinction between fully subjective models and models incorporating explicitly some information. We review definitions and characterizations of ambiguity aversion in these models. We mention the challenges posed by some of the models presented. We end with a review of part of the experimental literature and applications of these models to economic settings.