Phase transformations are among the most intriguing and technologically useful phenomena in materials, particularly with regard to controlling microstructure. After a review of thermodynamics, this book has chapters on Brownian motion and the diffusion equation, diffusion in solids based on transition-state theory, spinodal decomposition, nucleation and growth, instabilities in solidification, and diffusionless transformations. Each chapter includes exercises whose solutions are available in a separate manual. This book is based on the notes from a graduate course taught in the Centre for Doctoral Training in the Theory and Simulation of Materials. The course was attended by students with undergraduate degrees in physics, mathematics, chemistry, materials science, and engineering. The notes from this course, and this book, were written to accommodate these diverse backgrounds.