This article reviews various models for both discrete-time and continuous-time closed capture-recapture experiments. The traditional discrete-time models assume that the samples are independent. Dependence may be caused by local dependence (list dependence) within each animal or by heterogeneity among animals. Three different approaches that can incorporate dependence into models are reviewed, i.e., ecological models, log-linear models, and the sample-coverage approach, The statistical tools involved in population size estimation in these three approaches cover a wide range of methodologies. There has been relatively little published research for the continuous-time counterparts. The counting process approach, which is the framework for most existing estimation procedures for continuous-time models, is reviewed. The connection of continuous-time models to recurrent event analysis in the context of failure time inferences is discussed. The applications of capture-recapture models to other disciplines are briefly presented. Remarks about the limitations of the models are made and some future research directions are also suggested.