Secure Multi-Party Computation (SMPC) is a generic cryptographic primitive that enables distributed parties to jointly compute an arbitrary functionality without revealing their own private inputs and outputs. Since Yao’s seminal work in 1982, 30 years of research on SMPC has been conducted, proceeding from pure theoretical research into real-world applications. Recently, the increasing prevalence of the newly emerging technologies such as cloud computing, mobile computing and the Internet of Thing has resulted in a re-birth of SMPC’s popularity. This has occurred mainly because, as a generic tool for computing on private data, SMPC has a natural advantage in solving security and privacy issues in these areas. Accordingly, many application-oriented SMPC protocols have been constructed. This paper presents a comprehensive survey on the theoretical and practical aspects of SMPC protocols. Specifically, we start by demonstrating the underlying concepts of SMPC, including its security requirements and basic construction techniques. Then, we present the research advances regarding construction techniques for generic SMPC protocols, and also the cutting-edge approaches to cloud-assisted SMPC protocols. Then, we summarize the concrete application-oriented protocols that are currently available, and finally, we present a discussion of the current literature and conclude this survey.