The Simplex algorithm, developed by George B. Dantzig in 1947 represents a quantum leap in the ability of applied scientists to solve complicated linear optimization problems. Subsequently, its utility in solving finite models, including applications in transportation, production planning, and scheduling, have made the algorithm an indispensable tool to many operations researchers. This thesis is primarily an exploration of the simplex algorithm, and a discussion of the utility of the algorithm in unconventional optimization problems. The mathematical theory upon which the algorithm is based and a general description of the algorithm are presented. The reader is assumed to have little exposure to convexity, duality, or the Simplex algorithm itself. More important to the thesis are the examples that accompany the discussion of the Simplex algorithm. Herein are a variety of unusual applications for the algorithm, including applications in infinite dimensional vector spaces, uniform approximation, and computer assisted tomographic image reconstruction. These examples serve both to facilitate a better understanding of the algorithm, and to present it in unusual settings.