The field of multi-criteria optimization is reviewed as it pertains to lexicographic optimization over real-valued vector spaces. How lexicographic optimization differs from multi-criteria optimization that is restricted to proper Pareto optima is explained. Through a survey of previous work, it is revealed that there are currently no generally applicable methods for solving lexicographic optimization problems, and it is explained that this is due to the lack of an adequate mathematical theory for such problems. A more adequate mathematical theory is then presented for lexicographic optimization in this paper.