Source code summaries can help software developers comprehend programs faster and better, and assist maintenance developers in accomplishing their tasks efficiently. Since writing summaries by programmers is of high cost and low efficiency, researchers have tried to summarize source code automatically. In recent years, the technologies of neural network-based automatic summarization of source code have become the mainstream techniques of automatic source code summarization, and it is a hot research topic in the domain of intelligent software engineering. Firstly, this paper describes the concept of source code summarization and the definition of automatic source code summarization, presents its development history, and reviews the methods and metrics of the quality evaluation of the generated summaries. Then, it analyzes the general framework and the main challenges of neural network-based automatic code summarization algorithms. In addition, it focuses on the classification of representative algorithms, the design principle, characteristics, and restrictions of each category of algorithms. Finally, it discusses and looks forward to the trends on techniques of neural network-based source code summarization in future.