As the title indicates, this is a survey of results involving various types of closure conditions, where closure conditions involve adding an edge to a graph without changing the existence of some specified graphical property. Most of the graphical properties studied are hamiltonian type properties such as hamiltonian, panconnected, pancyclic, and cycle extendable, but other properties such as the existence of matchings are also studied. The Bondy-Chvátal closure, which started this line of investigation, is surveyed extensively along with the corresponding stability conditions. Results involving graphs with complete closures are also presented, as well as other related closure conditions such as the triple closure, \(0\)-dual closure, and neighborhood closure. For claw-free graphs, Ryjáček introduced a closure obtained by completing the neighborhoods of vertices with connected neighborhoods. Results using this closure are also described. This is a broad survey of ``closure'' results with an extensive list of references.