This paper introduces the basic idea and provides the mathematical formulation of the delayed feedback control (DFC) methodology, which has been widely used in chaos control. Stability analysis including the well_known odd number limitation of the DFC is reviewed. Some new developments in characterizing the limitation of the DFC are presented. Various modified DFC methods, which are developed in order to overcome the odd number limitation, are also described. Finally, some open problems in this research field are discussed. This paper introduces the basic idea and provides the mathematical formulation of the delayed feedback control (DFC) methodology, which has been widely used in chaos control. Stability analysis including the well_known odd number limitation of the DFC is reviewed. Some new developments in characterizing the limitation of the DFC are presented. Various modified DFC methods, which are developed in order to overcome the odd number limitation, are also described. Finally, some open problems in this research field are discussed.