Modern software systems usually contain enormous code comments which provide abundant information that have been leveraged to help perform various software engineering tasks, such as bug detection, specification inference, and code synthesis. However, existing work barely leverages program analysis to systematically derive, refine, and propagate comments. For example, by propagation via program analysis, comments can be passed on to code entities that are not commented such that code bugs can be detected leveraging the propagated comments. Developers usually comment on different aspects of code elements like methods, and use comments to describe various contents, such as functionalities and properties. To more effectively utilize comments, a fined-grained and elaborated taxonomy of comments and a reliable classifier to automatically categorize a comment are needed. In this paper, we build a comprehensive taxonomy and propose using program analysis to propagate comments. We develop a prototype CPC, and evaluate it on 5 projects. The evaluation results demonstrate 41573 new comments can be derived by propagation from other code locations with 88% accuracy. Among them, we can derive precise functional comments for 87 native methods that have neither existing comments nor source code. Leveraging the propagated comments, we detect 37 new bugs in open source large projects, 30 of which have been confirmed and fixed by developers, and 304 defects in existing comments (by looking at inconsistencies between existing and propagated comments), including 12 incomplete comments and 292 wrong comments. This demonstrates the effectiveness of our approach. Our user study confirms propagated comments align well with existing comments in terms of quality.