The Earth system—comprising atmosphere, ocean, land, cyrosphere and biosphere—is an immensely complex system, involving processes and interactions on a wide range of space– and time–scales. To understand and predict the evolution of the Earth system is one of the greatest challenges of modern science, with success likely to bring enormous societal benefits. High–performance computing, along with the wealth of new observational data, is revolutionizing our ability to simulate the Earth system with computer models that link the different components of the system together. There are, however, considerable scientific and technical challenges to be overcome. This paper will consider four of them: complexity, spatial resolution, inherent uncertainty and time–scales. Meeting these challenges requires a significant increase in the power of high–performance computers. The benefits of being able to make reliable predictions about the evolution of the Earth system should, on their own, amply repay this investment.