Understanding the causes and consequences of, and devising countermeasures to, global warming is a profoundly complex problem. Network representations are sometimes the only way forward, and sometimes able to reduce the complexity of the original problem. Networks are both necessary and natural elements of climate science. Furthermore, networks form a mathematical foundation for a multitude of computational and analytical techniques. We are only beginning to see the benefits of this connection between the sciences of climate change and network science. In this review, we cover the wide spectrum of network applications in the climate-change literature -- what they represent, how they are analyzed, and what insights they bring. We also discuss network data, tools, and problems yet to be explored.