Autonomous systems are emerging in many application domains. With the recent advancements in artificial intelligence and machine learning, sensor technology, perception algorithms and robotics, scenarios previously requiring strong human involvement can be handled by autonomous systems. With the independence from human control, cybersecurity of such systems becomes even more critical as no human intervention in case of undesired behavior is possible. In this context, this paper discusses emerging security challenges in autonomous systems design which arise in many domains such as autonomous incident response, risk assessment, data availability, systems interaction, trustworthiness, updatability, access control, as well as the reliability and explainability of machine learning methods. In all these areas, this paper thoroughly discusses the state of the art, identifies emerging security challenges and proposes research directions to address these challenges for developing secure autonomous systems.