In the classical problem of quickest change detection, a decision maker observes a sequence of random variables. At some point in time, the distribution of the random variables changes abruptly. The objective is to detect this change in distribution with minimum possible delay, subject to a constraint on the false alarm rate. In many applications of quickest change detection, e.g., where the changes are infrequent, it is of interest to control the cost of observations or the cost of data acquired before the change point. To this end, in this survey paper, data-efficient versions of the classical quickest change detection problems are studied, and a summary of existing results on these problems is provided. Some extensions to distributed sensor networks are also discussed.