Multi-stage robust optimization, in which decisions are taken sequentially as new information becomes available about the uncertain problem parameters, is a very versatile yet computationally challenging paradigm for decision-making under uncertainty. In this paper, we propose a new model and solution approach for multi-stage robust mixed-integer programs, which may contain both continuous and discrete decisions in any time stage. Our model builds upon the finite adaptability scheme developed for two-stage robust optimization problems, and it allows us to decompose the multi-stage problem into a large number of much simpler two-stage problems. We discuss how these two-stage problems can be solved both exactly and approximately, and we report numerical results for route planning and location-transportation problems.