With the intensive research and application of cloud computing, the problems about the security and privacy protection of the data stored on the remote server become hot issues of business and academia. Traditional cryptography methods can partly solve the above problems, but bring lots of obstacles to the remote user for data query and usage as well. Searchable encryption is an option to solve this contradiction, because of which enables users to search encrypted data stored on remote server, and has become a research hotspot in the field of information security at present. An overview of the public key searchable encryption was made in this paper, of which especially focused on the origin, background and research results in recent years of the Public key Encryption with Keyword Search (PEKS). The PEKS formal definition and security model definitions were given, followed by the detail discussion about the other aspects of this domain, such as the PEKS secure channel dependency problem, the query functional improvement, et al. Finally, the future trends, hotspot and open-ended problems of this novel research area were put forward.