Vehicle platooning has been studied for several decades, with objectives such as improved traffic throughput on existing infrastructure or reduced energy consumption. All the time, it has been apparent that safety is an important issue. However, there are no comprehensive analyses of what is needed to achieve safety in platooning, but only scattered pieces of information. This paper investigates, through a systematic literature review, what is known about safety for platooning, including what analysis methods have been used, what hazards and failures have been identified, and solution elements that have been proposed to improve safety. Based on this, a gap analysis is performed to identify outstanding questions that need to be addressed in future research. These include dealing with a business ecosystem of actors that cooperate and compete around platooning, refining safety analysis methods to make them suitable for systems-of-systems, dealing with variability in vehicles, and finding solutions to various human factors issues.