This article summarizes the fundamentals of radar polarimetry and gives examples of related applications and research. The mathematical foundation is described first. This discussion includes explanations of the various ways in which polarized and partially polarized electromagnetic waves can be described and how formalisms are used to express the polarimetric scattering properties of objects and surfaces. The scattering, Mueller, covariance, and coherency matrices embody these formalisms. Some practical aspects of polarimetric radar design are then discussed, and the article concludes by summarizing applications of polarimetry and areas of active research, including scattering studies, image classification, contrast enhancement, meteorology, and topographic mapping.