Many models for the spread of infectious diseases in populations have been analyzed math- ematically and applied to specific diseases. Threshold theorems involving the basic repro- duction number R0, the contact number σ, and the replacement number R are reviewed for the classic SIR epidemic and endemic models. Similar results with new expressions for R0 are obtained for MSEIR and SEIR endemic models with either continuous age or age groups. Values of R0 and σ are estimated for various diseases including measles in Niger and pertussis in the United States. Previous models with age structure, heterogeneity, and spatial structure are surveyed. 1. Introduction. The effectiveness of improved sanitation, antibiotics, and vac- cination programs created a confidence in the 1960s that infectious diseases would soon be eliminated. Consequently, chronic diseases such as cardiovascular disease and cancer received more attention in the United States and industrialized countries. But infectious diseases have continued to be the major causes of suffering and mortality in developing countries. Moreover, infectious disease agents adapt and evolve, so that new infectious diseases have emerged and some existing diseases have reemerged (142). Newly identified diseases include Lyme disease (1975), Legionnaire's disease (1976), toxic-shock syndrome (1978), hepatitis C (1989), hepatitis E (1990), and hantavirus (1993). The human immunodeficiency virus (HIV), which is the etiological agent for acquired immunodeficiency syndrome (AIDS), emerged in 1981 and has become an important sexually transmitted disease throughout the world. Antibiotic-resistant strains of tuberculosis, pneumonia, and gonorrhea have evolved. Malaria, dengue, and yellow fever have reemerged and are spreading into new regions as climate changes occur. Diseases such as plague, cholera, and hemorrhagic fevers (Bolivian, Ebola, Lassa, Marburg, etc.) continue to erupt occasionally. Surprisingly, new infectious agents called prions have recently joined the previously known agents: viruses, bac- teria, protozoa, and helminths (worms). There is strong evidence that prions are the cause of spongiform encephalopathies, e.g., bovine spongiform encephalopathy (BSE, "mad cow disease"), Creutzfeldt-Jakob disease (CJD), kuru, and scrapie in sheep (168). Recent popular books have given us exciting accounts of the emergence and de- tection of new diseases (82, 168, 170, 183). It is clear that human or animal invasions