Methods are described for the prediction of queue lengths and vehicular delays at road junctions subject to time-varying traffic demand and capacity. The properties of individual traffic streams are developed firstly in terms of the probability distribution of queue lengths, and scondly by approximation formulae which allow queues and delays to be predicted directly. Two approaches are used: (i) a high definition representation in which the queue length is developed in a sequence of small consecutive time intervals (five-minute intervals, for example), and (ii) a low definition representation which allows the overall delay for a peak period to be predicted given traffic flows averaged over a longer period ( an hour, for example). The first has applications in engineering design, network analysis, and similar problems and the second in the economic assessment of junction performance. The queueing characteristics of whole junctions are built up from those of the component streams. Major/minor priority junctions, roundabouts and traffic signals are considered.