Recall that the Lusternik-Schnirelmann category of a given topological space \(X\), LS-cat\((X)\), is the least integer \(n\) for which \(X\) can be covered by \(n+1\) open sets, each of which is contractible within \(X\). This numerical homotopy invariant, defined originally by Lusternik and Schnirelmann in 1934 has proven to be of great use in different branches of mathematics. This paper is a survey on the topological and geometrical properties of many LS invariants, all of them related to the original one. Beginning with Crit\((f)\), the number of critical points of a given function on a manifold, the author introduces and states the main results and properties (from the classical ones to even the most recent in which the author has played an important role) concerning cone length (of different kinds), strong LS-cat, cup-length, Toomer invariant, LS-category weights, stable versions of LS-cat, Ganea’s conjectures on LS-invariants, higher Hopf invariants, etc. The author also includes a section with open problems on these invariants and a fairly big list of spaces and/or manifolds for which some of these invariants have been computed.