I will present probabilistic proofs of some regularity properties for the value function of general optimal stopping problems and for the associated optimal boundaries. In particular this talk focusses on C^1 regularity of the value function and Lipschitz continuity of the optimal boundary. Most of our arguments rely on fundamental concepts from the theory of Markov processes and bridge the probabilistic and the analytical strands of the literature on free boundary problems. I will also illustrate situations in which our work improves or complements known facts from PDE theory.