Abstract: Cut-and-Choose is a widely used technique in cryptography, which plays an important role in the design of secure multi-party computation (MPC) protocols. The main idea of Cut-and-Choose is that one party constructs multiple copies of the objective garbled circuit, and the other party randomly chooses some of the circuits to be opened for correctness check. If the check passes, the parties evaluate the remaining circuits and determine the final output of the computation task. In the early research works, Cut-and-Choose was mainly used in MPC in malicious model, and a lot of excellent research results were proposed. Although this technology was also applied in the covert security model, it did not attract much attention at that time. In recent years, with in-depth research on covert adversaries, Cut-and-Choose technique and ideas based on this technique have also been used to obtain publicly verifiable covert secure MPC protocols. Some representative research works have emerged. In this work, we summarize the main research advance of Cut-and-Choose in malicious security model and covert security model, and present the achievements of this technology in the publicly verifiable covert security model. We make a detailed summary and analysis of relevant results in this research field and point out possible research directions in the future.