Morphological pattern formation and its spatio‐temporal evolution are among the most intriguing natural phenomena governed by nonlinear complex dynamics. These phenomena form the major study subjects of many fields including biology, hydrodynamics, chemistry, optics, materials science, ecology, geology, geomorphology, and cosmology, to name a few. In materials science, the morphological pattern is referred to as microstructure, which is characterized by the size, shape, and spatial arrangement of phases, grains, orientation variants, ferroelastic/ferroelectric/ferromagnetic domains, and/or other structural defects. These structural features usually have an intermediate mesoscopic length scale in the range of nanometers to microns. Microstructure plays a critical role in determining the physical properties and mechanical behavior of materials. Today the primary task of material design and manufacture is to optimize microstructures— by adjusting alloy composition and varying processing sequences—to obtain the desired properties. Microstructural evolution is a kinetic process that involves the appearance and disappearance of various transient and metastable phases and morphologies. The optimal properties of materials are almost always associated with one of these nonequilibrium states, which is “frozen” at the application temperatures. Theoretical characterization of the transient and metastable states represents a typical nonequilibrium nonlinear and nonlocal multiparticle problem. This problem generally resists any analytical solution except in a few extremely simple cases. With the advent of and easy access to high‐speed digital computers, computer modeling is playing an increasingly important role in the fundamental understanding of the mechanisms underlying microstructural evolution. In this article, methods recently developed for simulating the formation and dynamic evolution of complex microstructural patterns are reviewed. First, a brief account of the basic features of each method, including the conventional front‐tracking method (also referred to as the sharp‐interface method) and the new techniques without front‐tracking are given. Detailed description of the simulation technique and its practical application focuses on the field method since it is the most familiar and it is a more flexible method with broader applications. The need to link the mesoscopic microstructural simulation with atomistic calculation as well as with macroscopic process modeling and property calculation is also briefly discussed.

The key to predict material properties is the state of microstructure. Because microstructural evolution is a typical nonlinear, nonlocal, and multiparticle dynamic problem, computer simulations play an ever increasingly important role in predicting key microstructural features and their time evolution during various processes such as phase transformations, domain coarsening, and plastic deformation. A plethora of computational methods and algorithms have been developed in recent years to complement theoretical and experimental studies to explore the high dimensional material‐ and processing‐parameter space, which would not only lower the cost but also increase the efficiency of optimizing existing materials and developing new ones. In this article, a brief account of the basic features of each method, including the conventional front‐tracking methods and techniques without front‐tracking (such as Continuum/Microscopic/Coarse‐Grained Phase Field Method; Mesoscopic/Atomistic Monte Carlo; Cellular Automata; Discrete Lattice Model; Phase Field Crystal Model; Diffusive Molecular Dynamics; Microscopic Master Equations; Inhomogeneous Path Probability Method; and Molecular Dynamics) will be presented first, followed by detailed descriptions of the fundamentals of various phase‐field methods at different length scales and their model formulations. The applications of the phase field methods will be demonstrated by examples of coherent precipitation, grain growth, ferroelectric domain structure formation, dislocation core structures, and dislocation‐precipitate interactions. Existing challenges and future trends of the phase‐field methods are discussed at the end of the article.