Explore the intricacies of free boundary problems in this advanced mathematics lecture, focusing on the regularity of free boundaries in obstacle problems. Delve into the mathematical challenges of understanding free boundaries using methods from PDE, Calculus of Variations, and Geometric Measure Theory. Learn about the Stefan problem and the obstacle problem as classical examples in the field. Discover the types of possible blowups, classification of blowups, and the concept of free boundary. Examine the boundary harmonic inequality and its proof, as well as the study of singular points. Investigate the monotonicity formula and the proof of uniqueness of blowups. Conclude with an exploration of the continuous dependence on blowups, gaining valuable insights into current research and open problems in this fascinating area of mathematics.