Mobility is a basic daily necessity that is undergoing a major transformation driven both by societal needs and by technological developments. There are requirements to provide broader accessibility, to contribute to sustainable mobility solutions, to improve safety, etc. In addition, progress in a range of areas, such as sensing, algorithms for perception, verifiable decision-making, computing power, and communications technologies, together with more efficient machine learning methods, are enabling the development of autonomous systems capable of interacting with complex and dynamic environments. Over recent years, we have witnessed an increasing interest in the deployment of autonomous vehicles across land, sea and air demonstrating their potential to transport people and goods as well as providing solutions to a wide range of field applications. However, mobile autonomous systems with high levels of autonomy remain scarce, with only a few and each predominantly operating under very constrained conditions or relying on specialist operators for safety assurance. Deploying truly autonomous systems has been found to be more difficult than expected, for example, autonomous passenger vehicles. The acceptability of autonomous mobile platforms depends on our ability to demonstrate their safe, reliable, and continuous operation. One of the major concerns is safety when operating in open or public spaces; providing evidence to all stake-holders of the safe behaviour of mobile systems in these situations remains a challenge. This is compounded by the need to interact with other traffic agents whose behaviour might well be unpredictable. Experience has shown that there can be a very large number of scenarios that these systems need to address even in well-defined operational design domains. Extensive testing in real conditions, though enabling some progress, will be insufficient to address all these situations. Meanwhile, society might be particularly unforgiving of failures, compromising long term progress across the area. Considerable work has been carried out, both in industry and academia, towards demonstrating how assurance in such domains might be achieved, as we strive to deploy autonomous mobile systems in public spaces independent of human control. This includes the development of related standards. However, there clearly remain significant steps that must be taken. Some of these may concern research that is needed, some concern practical development that needs to be undertaken, some concern publicising existing solutions and educating a range of stakeholders, etc. But what are these steps? Where are the gaps? And which developments are the most important?