There is a rapidly advancing literature on the macroeconomics of climate change. This review focuses on developments in the construction and solution of structural integrated assessment models (IAMs), highlighting the marriage of state-of-the-art natural science with general equilibrium theory. We discuss challenges in solving dynamic stochastic IAMs with sharp nonlinearities, multiple regions, and multiple sources of risk. Key innovations in deep learning and other machine learning approaches overcome many computational challenges and enhance the accuracy and relevance of policy findings. We conclude with an overview of recent applications of IAMs and key policy insights.