Despite advances m understanding the basic biology of HIV the aetiological agent of AIDS, medica1, public health and health education planning is plagued by uncertainties Mathematical models of the dynamics of HIV transmission and its progression to AIDS can clarify what data must be collected in order to predict future prevalence, make predictions about the likely effect of future intervention pobcies and provide predictions for several decades ahead. The motivation of this research is to provide reliable estimates of the incidence of HIV infection and AIDS in the Irish population. In Chapters 1 and 2 we discuss the background to the disease in Ireland and the role of mathematical modelling in the spread AIDS. From this we show where key epidemiological data is lacking and how models to date have concentrated on the spread of the disease within the homosexual population. In Chapter 3 we describe the adjustment of the number of AIDS cases to allow for reporting delays Subsequently we consider the solution of the integral equation models generated by the back-projection method for the adjusted AIDS cases. In Chapter 4 we improve upon the estimates of the incidence of HIV infection found in Chapter 3 by evaluating the integral arising in back-projection, in terms of a gamma function plus a remainder in the form of a series in t. We also provide error bounds for the remainder. This new solution allows us to predict new and more reliable estimates of the level of HIV infection m Ireland. In Chapter 5 we provide estimates of the minimum number of deaths from AIDS, based on the number of AIDS cases known to the Department of Health and the distribution of the length of survival times after the onset of AIDS. The results of a HIV transmission survey are presented in Chapter 6 These provide detailed information on the habits and behaviour of those at risk of HIV infection and allow us to derive preliminary model parameters. Finally in Chapter 7 we develop and implement a nonlinear deterministic differential equation model for the spread of HIV and its progression to AIDS m the Irish IVDU and homosexual populations. We examine the effects of likely intervention policies on the extent and spread of the disease and we make recommendations based on our thesis findings.