In epidemiologyR0denotes the average number of secondary cases of an infectious disease that one case would generate in a completely susceptible population. This concept is among the foremost and most valuable ideas that mathematical thinking has brought to epidemic theory. In this contribution, we first review the historical development of Ro, from demography to epidemiology, proceed to give an exposition of the recently formalised theory to define and calculateR0for structured populations, return to the interaction of demography and epidemiology for an example of the use of the concept to study vaccination campaigns and finally we deal with statistical aspects of estimatingR0. In the appendix we discuss some issues of current attention.