Speech communication is fundamental to human society yet, as a result of the relative inaccessibility of the larynx and vocal tract to instrumentation and the complexity of the aerodynamics therein, the mechanism of sound production during speech has not been fully quantified. In this paper we consider vowel production modelling. We describe those aspects of the anatomy and physiology of the speech system that are relevant to the generation of vowel sounds and outline the means by which the larynx is caused to vibrate. We then discuss the equations of fluid mechanics required to model the laryngeal airflow, describing the approximations commonly used to reduce them to a solvable set and assessing the validity of those approximations for speech air-flows during voicing. We next consider the sources of sound that generate the radiated vowel waveforms; we include the traditional glottal-waveform source and a number of other mechanisms that may contribute to the output speech wave. Finally we outline the difficulties in obtaining data from live subjects and some of the methods used to overcome these difficulties