Robotic research is making huge progress. However, existing solutions are facing a number of challenges preventing them from being used in our everyday tasks: (i) robots operate in unknown environments, (ii) robots collaborate with each other and even with humans, and (iii) robots shall never injure people or create damages. Researchers are targeting those challenges from various perspectives, producing a fragmented research landscape. We aim at providing a comprehensive and replicable picture of the state of the art from a software engineering perspective on existing solutions aiming at managing safety for mobile robotic systems. We apply the systematic mapping methodology on an initial set of 1,274 potentially relevant research papers, we selected 58 primary studies and analyzed them according to a systematicallydefined classification framework. This work contributes with (i) a classification framework for methods or techniques for managing safety when dealing with the software of mobile robotic systems (MSRs), (ii) a map of current software methods or techniques for software safety for MRSs, (iii) an elaboration on emerging challenges and implications for future research, and (iv) a replication package for independent replication and verification of this study. Our results confirm that generally existing solutions are not yet ready to be used in everyday life. There is the need of turn-key solutions ready to deal with all the challenges mentioned above.