This review presents an historical account of the developments of the statistical analysis of quantal transmission over the past half century and of the progress made in using this approach to reveal new properties of nerve terminals. In the early 1950s, Katz and his colleagues showed that evoked transmitter release occurred in quanta at the neuromuscular junction, opening up the study of transmitter release at nerve terminals to statistical analysis. In the subsequent two decades attempts were made to see if evoked quantal release could be described by binomial or compound binomial statistics, as originally suggested by Katz, and to relate the parameters of the statistic to various structures of the nerve terminal. During this period two hypotheses were enunciated, namely the 'vesicle hypothesis', which states that quanta arise as a consequence of the packaging of transmitter in vesicles; and the 'active zone hypothesis', which states that vesicles undergo exocytosis at discrete sites on the nerve terminal. Unsuccessful attempts were made to relate the binomial parameter n to the elements in these hypotheses, that is to the number of active zones possessed by the terminal or the number of vesicles available for release at these zones. This difficulty was part resolved in the late 1970s with the application of non-uniform binomial statistics to transmitter release from nerve terminals, in which n is the number of active zones each with their individual probabilities, p(j). Autocorrelation functions were subsequently introduced to detect if transmitter release is quantised at a particular nerve terminal. Statistical methods which would allow discrimination between different models of transmitter release over the active zones of a terminal were then developed. The introduction of maximum likelihood estimation procedures then allowed estimates to be made of the parameters in the statistical models of quantal release. The application of these procedures to experimental data from a variety of nerve terminals provided evidence for the concept that each synapse, taken as possessing a single active zone, possesses its own individual probability of secretion of a quantum by the exocytosis of a vesicle. In the late 1960s Stevens introduced the first stochastic approach to the analysis of the kinetics of the release of a quantum of transmitter at the neuromuscular junction following an impulse. In the subsequent decades this was developed into an explicit theory for the interaction of proteins involved in regulated exocytosis of a vesicle at an active zone. The parameters were the number of transition steps in the release process (k), each occurring at the same rate (alpha), with the possibility of each of these steps becoming blocked at the same rate (gamma). Maximum likelihood estimation procedures could then be used to obtain these parameter values. The discovery was made in the 1990s of the core proteins of the SNARE complex that govern regulated exocytosis. This offers the possibility in the near future of identifying the kinetic interaction of these proteins with the parameters of the stochastic process of exocytosis which confer a particular probability on individual synapses.