Structural models have become a frontier tool in business and economics research. In this survey, we discuss the literature on structural models for the prescription drug market, which has attracted a significant attention from researchers in marketing and economics, and related fields. The literature has evolved from adopting standard structural models developed for other markets to models that are specifically designed to capture the nuance of drug prescription behaviour. Along the way, these empirical frameworks have not only greatly improved in explaining stylized facts, but also at producing better (more reasonable) counterfactual predictions. Topics covered by this survey include the application of learning models to explain slow diffusion, post-patent expiry competition, pre-patent expiry competition, R&D and new drug introduction, managerial and public policy analysis, and the economics of the Medicare Part-D program. We conclude by discussing future research directions.