This article presents and analyzes existing taxonomies of virtual and augmented reality and demonstrates knowledge gaps and mixed terminology, which may cause confusion among educators, researchers, and developers. Several such occasions of confusion are presented. A methodology is then presented to construct a taxonomy of virtual reality and augmented reality applications based on a combination of: a faceted analysis approach for the overall design of the taxonomy; an existing taxonomy of educational objectives to derive the educational purpose; an information systems analysis to establish important facets of the taxonomy; and two systematic mapping studies to identify categories within each facet. Using this methodology, a new taxonomy is proposed and the implications of its facets (and combinations of facets) are demonstrated. The taxonomy focuses on technology used to provide the virtual or augmented reality as well as the content presented to the user, including the type of gamification and how it is operated. It also accommodates a large number of devices and approaches developed throughout the years and for multiple industries, and provides a way to categorize them in order to clarify communication between researchers, developers, and educators. Use of the taxonomy is then demonstrated in two case studies--a virtual reality chemical plant for use in chemical engineering education and an augmented reality dog for veterinary education.