In the mathematic area of statistics and probability, Markov chains are a fundamental class of stochastic processes. Markov chains, like other mathematic tools, are extremely important to various fields of operational research, economics, finance, social sciences, computer science and engineering, networks, biology, physics, chemistry, and mechanics. The powerful Markov decision process algorithm in reinforcement learning, which will be introduced in Volume II of this book, is developed based on Markov chains. In materials science and engineering, the properties and performance of a material depend on its chemical composition and microstructure, which is in turn formed by the topological arrangement of all involved atoms. Provided that the chemical composition of a material does not change during its service life, the material performance depends only on its microstructure that varies along with service time. The microstructure of a material is named as the state in Markov chains, and the change in the microstructure can be described by the variation in the state. For example, the hot deformation behavior of metals depends on the change in the microstructure, which involves various crystalline defect activities including dislocation multiplication, movement, topological configuration, annihilation, grain boundary movement, phase transformation, dynamic recrystallization, etc. On one side, material researchers are endeavoring to in situ characterize or monitor the change in the microstructure during tests or service, and on the other side, are doing their best to model dynamic behaviors of materials, where Markov chains will find new applications.