A very fundamental problem in quantum statistical mechanics involves whether--and how--an isolated quantum system will thermalize at long times. In quantum systems that do thermalize, the long-time expectation value of any "reasonable" operator will match its predicted value in the canonical ensemble. The Eigenstate Thermalization Hypothesis (ETH) posits that this thermalization occurs at the level of each individual energy eigenstate; in fact, any single eigenstate in a microcanonical energy window will predict the expectation values of such operators exactly. In recent work, we have identified, for a generic model system, precisely which operators satisfy ETH, as well as the limits to the information contained in a single eigenstate. Remarkably, our results strongly suggest that a single eigenstate can contain information about energy densities--and therefore temperatures--far away from the energy density of the eigenstate. After considering eigenstates, I will return to the more general case of time evolution following a quantum quench, and study which operators thermalize for typical initial states. Possible experimental implications will be discussed.