I introduce a model of belief updating, Coarse Bayesian Updating, where a boundedly rational agent, upon receipt of new information, applies a set of subjective criteria to select among competing theories of the world. The agent is characterized by a prior, a partition of the space of all probability distributions, and a representative distribution for each cell of the partition. When new information arrives, the agent computes the Bayesian posterior, determines which cell of the partition it belongs to, and adopts the representative of that cell as his posterior belief. The model includes Bayesian updating as a special case and accommodates many documented violations of Bayesian updating, including both under- and over-reaction to information. I provide behavioral characterizations of this procedure and analyze how it relates to other models and evidence on non-Bayesian updating. I also characterize what it means for an agent to be more sophisticated, and how Coarse Bayesians value information. The model employs standard primitives and, therefore, can be applied in most economic environments.