The vacuum landscape of string theory can solve the cosmological constant problem, explaining why the energy of empty space is observed to be at least 60 orders of magnitude smaller than several known contributions to it. It leads to a 'multiverse' in which every type of vacuum is produced infinitely many times, and of which we have observed but a tiny fraction. This conceptual revolution has raised tremendous challenges in particle physics and cosmology. To understand the low-energy physics we observe, and to test the theory, we will need novel statistical tools and effective theories. We must also solve a long-standing fundamental problem in cosmology: how to define probabilities in an infinite universe where every possible outcome, no matter how unlikely, will be realized infinitely many times. This 'measure problem' is inextricably tied to the quantitative prediction of the cosmological constant.