Cloud service mode has great economical and technical advantages and wide application prospects. The popularization of the cloud service is significant to both the informationization and the development of China. Cloud security is the most serious challenge in the generalization and the applications of the cloud service. Homomorphic encryption schemes, especially fully ones, are the most important technology to solve the security problem arising in cloud service, and a focus in the international cryptographic community. In this paper, we summarize the state of the art of the homomorphic encryption research, introduce the applications of the homomorphic encryption to the protection of the data confidentiality in cloud computing and to other fields, analyze the merits and the faults of various algebraic somewhat homomorphic encryption schemes and of fully homomorphic encryption schemes based on circuits, point out some open problems and new directions in the fully homomorphic encryption research, and briefly introduce the concept of secure plaintext computing, its advantages over cipher-text computing and some problems that need further studying.