AI-based systems are software systems with functionalities enabled by at least one AI component (e.g., for image-, speech-recognition, and autonomous driving). AI-based systems are becoming pervasive in society due to advances in AI. However, there is limited synthesized knowledge on Software Engineering (SE) approaches for building, operating, and maintaining AI-based systems. To collect and analyze state-of-the-art knowledge about SE for AI-based systems, we conducted a systematic mapping study. We considered 248 studies published between January 2010 and March 2020. SE for AI-based systems is an emerging research area, where more than 2/3 of the studies have been published since 2018. The most studied properties of AI-based systems are dependability and safety. We identified multiple SE approaches for AI-based systems, which we classified according to the SWEBOK areas. Studies related to software testing and software quality are very prevalent, while areas like software maintenance seem neglected. Data-related issues are the most recurrent challenges. Our results are valuable for: researchers, to quickly understand the state-of-the-art and learn which topics need more research; practitioners, to learn about the approaches and challenges that SE entails for AI-based systems; and, educators, to bridge the gap among SE and AI in their curricula.