The Lusternik-Schnirelmann (LS) category is a numerical invariant that measures the complexity of a topological space. Easy to define but difficult to compute, this invariant has given rise to many interesting and exciting problems. After giving an extremely selective survey of some results related to LS category, we discuss the categorical sequence, a convenient system to help compute LS category. We then introduce a "weighted" notion of cone-length, a closely related invariant to LS category. We prove a relationship between the weighted cone length and categorical sequences, showing as a corollary that all positive rational numbers can be realized as the weighted cone-length of some space.