As a feedback control methodology exclusively targeting asynchronous sequential machines (ASMs), corrective control has been rapidly developing for the past two decades. This paper presents a comprehensive survey on the theory and application of dynamic corrective control in which the controller also has the form of an ASM. First, basic notions and principles of dynamic corrective control, including models of ASMs and configurations of closed-loop systems, are reviewed. Next, assorted dynamic corrective control schemes are presented aiming at solving specific control problems of ASMs, such as model matching and fault-tolerant control. Variations of control aspects are classified according to modeling formalisms of controlled ASMs—input/state, input/output, and composite ASMs—and involved fault characteristics, such as transient, permanent, and intermittent faults and intelligent attacks. Representative results on the application of fault-tolerant corrective control to real-world engineering systems are also provided with an emphasis on space-borne digital systems. Finally, some challenging topics for future studies on corrective control are discussed.