Reinforcement learning (RL) has a rich history in neuroscience, from early work on dopamine as a reward prediction error signal (Schultz et al., 1997) to recent work proposing that the brain could implement a form of 'distributional reinforcement learning' popularized in machine learning (Dabney et al., 2020). There has been a close link between theoretical advances in reinforcement learning and neuroscience experiments throughout this literature, and the theories describing the experimental data have therefore become increasingly complex. Here, we provide an introduction and mathematical background to many of the methods that have been used in systems neroscience. We start with an overview of the RL problem and classical temporal difference algorithms, followed by a discussion of 'model-free', 'model-based', and intermediate RL algorithms. We then introduce deep reinforcement learning and discuss how this framework has led to new insights in neuroscience. This includes a particular focus on meta-reinforcement learning (Wang et al., 2018) and distributional RL (Dabney et al., 2020). Finally, we discuss potential shortcomings of the RL formalism for neuroscience and highlight open questions in the field. Code that implements the methods discussed and generates the figures is also provided.