A Review of Safe Online Learning for Nonlinear Control Systems**This work has been jointly funded by the EPSRC and BAE Systems under an Industrial CASE studentship. The authors would also like to thank the following researchers for their kind assistance. Sumeet Singh, Ian Manchester and Johan Löfberg.

Learning for autonomous dynamic control systems that can adapt to unforeseen environmental changes are of great interest but the realisation of a practical and safe online learning algorithm is incredibly challenging. This paper highlights some of the main approaches for safe online learning of stabilisable nonlinear control systems with a focus on safety certification for stability. We categorise a non-exhaustive list of salient techniques, with a focus on traditional control theory as opposed to reinforcement learning and approximate dynamic programming. This paper also aims to provide a simplified overview of techniques as an introduction to the field. It is the first paper to our knowledge that compares key attributes and advantages of each technique in one paper.