The study of similarities and dissimilarities between the blocking and anti-blocking polyhedra began with a series of celebrated papers by Fulkerson (1970, 1971, 1972), and it has grown up a mature theory by significant contributions of Lehman, Lovász, Padberg, and others in 1970s and 1980s. Even today, this theory still shows a big progression such as the perfect graph theorem of Seymour et al. (2006). In this paper, we survey the current status of this research field with a focus on the conjecture of Conforti & Cornuéjols and the conjecuture of Grinstead.