This book is one of the very first on the subject of mathematical materials science and presents a view different from that prevalent in the physical literature. Issues foundational in nature are stressed with the emphasis on the interplay between mathematics and physics. It discusses the dynamics of two-phase systems within the framework of modern continuum thermodynamics. Two general theories are considered and the resulting equations exhibit unstable growth patterns. The free boundary problems that form the basis of the subject should be of great interest to mathematicians and physical scientists.