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Real-world autonomous systems are often controlled using conventional model-based control methods. But if accurate models of a system are not available, these methods may be unsuitable. For many safety-critical systems, such as robotic systems, a model of the system and a control strategy may be learned using data. When applying learning to safety-critical systems, guaranteeing safety during learning as well as testing/deployment is paramount. A variety of different approaches for ensuring safety exists, but the published works are cluttered and there are few reviews that compare the latest approaches. This paper reviews two promising approaches on guaranteeing safety for learning-based robust control of uncertain dynamical systems, which are based on control barrier functions and control Lyapunov functions. While control barrier functions provide an option to incorporate safety in terms of constraint satisfaction, control Lyapunov functions are used to define safety in terms of stability. This review categorises learning-based methods that use control barrier functions and control Lyapunov functions into three groups, namely reinforcement learning, online and offline supervised learning. Finally, the paper presents a discussion of the suitability of the different methods for different applications.