The testing of drug dissolution rates from solid dosage forms is a very important area of research within the pharmaceutical industry. The ability to produce drugs with a given dissolution rate will lead to improved performance in the treatment of patients and will be of economic benefit to the pharmaceutical industry. However, dissolution testing in laboratories, aimed at reflecting in-vivo conditions, can be both time consuming and costly. Currently, most simulations of drug dissolution take place in standardized USP (United States Pharmaceutical) apparatuses. A number of these apparatuses exist, and it is the aim of this thesis to analyse drug dissolution in both the USP Paddle Apparatus and the USP Flow Through Apparatus. The first part of this thesis examines drug dissolution from a solid dosage form (compact) in the USP Paddle Apparatus. The process is set up as a boundary layer problem for which there exists both a momentum boundary layer and a concentration boundary layer. The dominant mass transfer mechanism is that of forced convection. A semi-analytical technique is used to solve the boundary layer equations for which velocity data has been provided from computational fluid dynamic simulations. Wherever possible the results from this semi-analytical approach have been compared with that of an exact solution. The second part of the thesis concentrates on the USP Flow Through Apparatus. As the process of drug dissolution in the Flow Through Apparatus is dependent on a vertical flow, the analysis is complicated by the introduction of buoyancy effects. Chapters five to nine analyse a number of general cases for buoyancy driven flows on both flat and curved surfaces. Later, in chapter ten, these general cases are then applied to the process of drug dissolution from the surface of a compact in the USP Flow Through Apparatus. Throughout the thesis, the predicted dissolution rates from the theoretical approach are compared with those of experiment