Linear programming is undeniably a central tool of applied mathematics and a source of many fascinating mathematical problems. In this talk I will present several advances from the past 5 years in the theory of linear optimization. These results include new results on the complexity of the simplex method, the structure of central paths of interior point methods, and about the geometry of some less well-known iterative techniques. I will try to summarize work by many authors and will include results that are my own joint work with subsets of the following people A. Basu, M. Junod, S. Klee, B. Sturmfels, and C. Vinzant.