Refinery complexity quantifies the sophistication and capital intensity of a refinery and has found widespread application in facility classification, cost estimation, sales price models, and other uses. Despite the ubiquity and widespread use of refining complexity, however, surprisingly little material has been written on its applications. The purpose of this review is to describe the primary applications of refinery complexity and some recent extensions. A secondary purpose of this review is to provide a framework that unifies complexity applications and suggests avenues for future research. Examples illustrate the applications considered.