Chronic diseases and pathological medical conditions requiring the administration of longterm pharmaceutical dosages have in the past been treated by oral administrations of tablets, pills and capsules or through the use of creams and ointments, suppositories, aerosols, and injectables. Such forms of drug delivery, which are still currently used today, provide a prompt release of the drug, but with significant fluctuations in the drug levels within various regions of the body. Repeated administrations of the drug are often needed, at rather precise intervals of time, in order to maintain these levels within a relatively narrow therapeutic range as a means of assuring effectiveness at the low end and of minimizing adverse effects at the higher end of the fluctuation spectrum. Recent technical advances now permit one to control the rate of drug delivery. The required therapeutic levels may thus be maintained over long periods of months and years through implanted rate-controlled drug release capsules. Two such novel drug delivery systems currently employed are implanted erodible polymeric and ceramic capsules. Mathematical modeling and computer simulations can be very effective in improving and optimizing the performance of the self-regulating release of therapeutic drugs into specific regions of the body. Further development is needed for the optimal design of such capsules. It is in this area, in particular, that a review will be presented of the mathematical modeling techniques susceptible to refine the development of a reliable tool for designing and predicting the resulting pharmaceutical dosages as a function of time and space. Of primary importance in such models are the time-varying effective permeability of the capsule to the various molecules composing the drug, the effective solubility and diffusion coefficients of the drug and its metabolites in the surrounding tissues and fluids and, finally, the uptake of the drug at the target organ. Mathematical models are presented for the diffusional release of a solute from an erodible matrix in which the initial drug loading c0 is greater than the solubility limit cs. An inward moving diffusional front separates the reservoir (unextracted region) containing the undissolved drug from the partially extracted region. The mathematical formulation of such moving boundary problems has wide application to heat transfer with melting phase transitions and diffusion-controlled growth of particles, in addition to our topic of controlled-release drug delivery. In spite of this diversity of applications, only a very few mathematical descriptions have been published for the analysis of release kinetics of a dispersed solute from polymeric or ceramic matrices. In these rare instances, perfect sink conditions are assumed, while matrix swelling, concentration-dependence of the solute diffusion coefficient and the external mass transfer resistance have been largely neglected. The ultimate goal of such an investigation is to provide a reliable design tool for the fabrication of specialized implantable capsule/drug combinations which will deliver pre-specified and reproducible dosages over a wide spectrum of conditions and required durations of therapeutic treatment. Such a mathematical/computational tool can also prove effective in the prediction of suitable dosages for other drugs of differing chemical and molecular properties which have not been subjected to time-consuming animal laboratory testing. Finally, such models may permit more realistic scaling of the required dosages of therapeutic drug for variations in diverse factors such as body weight or organ size and capacity of the patient (clinical medicine) or animal (veterinary medicine for farm animals). Additional applications of controlled-release drug delivery for insecticide and pesticide use in agriculture, and the control of pollution in lakes, rivers, marshes, etc. in which a pre-programmed dose-time schedule is necessary, further