This article reviews the theory and applications of the Tensor LEED approximation in surface crystallography by low-energy electron diffraction (LEED). The motivation for the development of the approximation is explored by reviewing the limitations of previous approaches to surface structure determination by LEED. The theory of the approximation is presented and its validity explored. The application of the approximation to automated search strategies, and disordered surfaces is discussed. The use of Tensor LEED theory as the basis of direct methods in LEED is reviewed. The article concludes with an evaluation of future prospects and applications of the theory of Tensor LEED.