Polymer dissolution is an important phenomenon in polymer science and engineering that has found applications in areas like microlithography, controlled drug delivery, and plastics recycling. This review focuses on the modeling efforts to understand the physics of the drug release process from dissolving polymers. A brief review of the experimentally observed dissolution behavior is presented, thus motivating the modeling of the mechanism of dissolution. The main modeling contributions have been classified into two broad approaches - phenomenological models and Fickian equations, and anomalous transport models and scaling law-based approaches. The underlying principles and the important features of each approach are discussed. Details of the important models and their corresponding predictions are provided. Experimental results seem to be qualitatively consistent with the present picture.