Building sensible processing-structure-property (PSP) links to gain fundamental insights and understanding of materials behavior has been the focus of many works in computational materials science. Microstructure characterization and reconstruction (MCR), coupled with machine learning techniques and materials modeling and simulation, is an important component of discovering PSP relations and inverse material design in the era of high-throughput computational materials science. In this article, we provide a comprehensive review of representative approaches for MCR and elaborate on their algorithmic details, computational costs, and how they fit into the PSP mapping problems. Multiple categories of MCR methods relying on statistical functions (such as n-point correlation functions), physical descriptors, spectral density function, texture synthesis, and supervised/unsupervised learning are reviewed. As no MCR method is applicable to the analysis and (inverse) design of all material systems, our goal is to provide the scientific community with a close examination of the state-of-the-art techniques for MCR, as well as useful guidance on which MCR method to choose and how to systematically apply it to a problem at hand. We illustrate applications of MCR on materials modeling and building structure property relations via two examples: One on learning the materials law of a class of composite microstructures, and the second on relating the permittivity and dielectric loss to a structural parameter in nanodielectrics. (C) 2018 Elsevier Ltd. All rights reserved.