  This paper surveys recent numerical advances in the phase field method for
geometric surface evolution and related geometric nonlinear partial
differential equations (PDEs). Instead of describing technical details of
various numerical methods and their analyses, the paper presents a holistic
overview about the main ideas of phase field modeling, its mathematical
foundation, and relationships between the phase field formalism and other
mathematical formalisms for geometric moving interface problems, as well as the
current state-of-the-art of numerical approximations of various phase field
models with an emphasis on discussing the main ideas of numerical analysis
techniques. The paper also reviews recent development on adaptive grid methods
and various applications of the phase field modeling and their numerical
methods in materials science, fluid mechanics, biology and image science.
