Delve into the fourth lecture of a series on free boundary problems in obstacle problems, exploring advanced concepts in partial differential equations, calculus of variations, and geometric measure theory. Examine the regularity of free boundaries, a key challenge in understanding solutions to problems that arise in physics, geometry, probability, biology, and finance. Learn about the Stefan problem and the obstacle problem as classical examples in free boundary studies. Gain insights into current research, open problems, and proven results in this field, building upon the foundations laid in previous lectures.