Explore the fascinating world of free boundary problems in this comprehensive lecture on the regularity of free boundaries in obstacle problems. Delve into the mathematical challenges of understanding free boundaries, which appear in various fields such as Physics, Geometry, Probability, Biology, and Finance. Learn about the Stefan problem and the obstacle problem, two classical examples in the study of free boundary problems. Discover the main known results, current research, and open problems in this field. Gain insights into the regularity theory of minimal surfaces, optimal regularity, and the regularity strategy. Examine proofs, blowups, monotonicity, and common properties of free boundaries. Understand the classification of blowups and their significance in the study of obstacle problems.