Although artificial intelligence (AI) is solving real-world challenges and transforming industries, there are serious concerns about its ability to behave and make decisions in a responsible way. Responsible AI has become a major issue in society. Many ethical regulations, principles, and guidelines for responsible AI have been issued by governments, research organizations, and enterprises in recent years. However, these principles are very high level and difficult to put into practice. On the other hand, most AI researchers focus on algorithmic solutions in data and models, while the responsible AI challenges actually crosscut the engineering lifecycle and all components of AI systems. To close the gap in operationalizing responsible AI, this paper aims to develop a research roadmap on software engineering for responsible AI. The proposed roadmap focuses on (i) establishing multi-level governance for responsible AI systems, (ii) setting up the development processes incorporating process-oriented best practices for responsible AI systems, and (iii) building responsible-AI-by-design into AI systems through system-level architectural style, patterns and techniques.