Free boundary problems are a category of mathematical problems characterised by the determination of both the solution to a partial differential equation and the dynamic boundary where the solution is defined. These problems are prevalent in various scientific and engineering contexts, including fluid dynamics, materials science and biological growth processes, where the boundary conditions are not predetermined but instead form part of the solution itself. Regularity theory examines the smoothness and structural properties of the solutions and their boundaries. Regularity results are essential for ensuring that solutions to free boundary problems are not only mathematically robust but also physically realistic. Free boundary problems and regularity theory are fundamental in fields requiring precise modelling of dynamic interfaces, thereby enhancing the predictive accuracy of scientific and engineering models. This book presents the complex subject of mathematical analysis in the most comprehensible and easy to understand language. While understanding the long-term perspectives of the topics, the book makes an effort in highlighting their impact as a modern tool for the growth of the discipline. Through this book, we attempt to further enlighten the readers about the new concepts in this field.