This paper reviews the historical background and the theoretical development of models for the analysis of data from capture-recapture or multiple-record systems for estimating the size of closed populations. The models and methods were originally developed for use in fisheries and wildlife biology and were later adapted for use in connection with human populations. Application to epidemiology came much later. The simplest capture-recapture model involves two lists or samples and has four key assumptions: that the population is closed, that individuals can be matched from capture to recapture, that capture in the second sample is independent of capture in the first sample, and that the capture probabilities are homogeneous across all individuals in the population. Log-linear models provide a convenient representation for this basic capture-recapture model and its extensions to K lists. The paper provides an overview for these models and illustrates how they allow for dependency among the lists and heterogeneity in the population. The use of log-linear models for estimation in the presence of both dependence and heterogeneity is illustrated on a four-list example involving ascertainment of diabetes using data gathered in 1988 from residents of Casale Monferrato, Italy. The final section of the paper discusses techniques for model selection in the context of models for estimating the size of populations.