Backpropagation (BP) is the standard algorithm for training the deep neural networks that power modern artificial intelligence including large language models. However, BP is energy inefficient and unlikely to be implemented by the brain. This thesis studies an alternative, potentially more efficient brain-inspired algorithm called predictive coding (PC). Unlike BP, PC networks (PCNs) perform inference by iterative equilibration of neuron activities before learning or weight updates. Recent work has suggested that this iterative inference procedure provides a range of benefits over BP, such as faster training. However, these advantages have not been consistently observed, the inference and learning dynamics of PCNs are still poorly understood, and deep PCNs remain practically untrainable. Here, we make significant progress towards scaling PCNs by taking a theoretical approach grounded in optimisation theory. First, we show that the learning dynamics of PC can be understood as an approximate trust-region method using second-order information, despite explicitly using only first-order local updates. Second, going beyond this approximation, we show that PC can in principle make use of arbitrarily higher-order information, such that for feedforward networks the effective landscape on which PC learns is far more benign and robust to vanishing gradients than the (mean squared error) loss landscape. Third, motivated by a study of the inference dynamics of PCNs, we propose a new parameterisation called "$μ$PC", which for the first time allows stable training of 100+ layer networks with little tuning and competitive performance on simple tasks. Overall, this thesis significantly advances our fundamental understanding of the inference and learning dynamics of PCNs, while highlighting the need for future research to focus on hardware co-design if PC is to compete with BP at scale.