The conditional intensity function of a spatial point process describes how the probability that a point of the process occurs at a particular point in its carrier space depends on the realisation of the process in the remainder of the carrier space. Provided that the point process is simple, the conditional intensity determines all of the properties of the process, in particular its likelihood function. In this paper we review the use of the conditional intensity function in the formulation of point process models and in making inferences from point process data, giving separate consideration to temporal, spatial and spatio-temporal settings. We argue that the conditional intensity function should take centre-stage in spatio-temporal point process modelling and analysis.