The (one-phase) Bernoulli problem deeply influenced the modern free boundary regularity theory and is still an object of intensive research. Free boundary problems are often PDE-described problems with a priori unknown (free) interfaces or boundaries. Such types of problems appear in Physics, Geometry, Probability, Biology, or Finance, and the study of solutions and free boundaries uses methods from PDE, Calculus of Variations, and Geometric Measure Theory. We first review the very classical theory, which is about how to understand the regularity of free boundaries. Then, we present some recent results for the Bernoulli problem with the p-Laplace operator.