The cornea of the human eye acts as a mirror that reflects light from a person's environment. These corneal reflections can be extracted from an image of the eye by modeling the eye-camera geometry as a catadioptric imaging system. As a result, one obtains the visual information of the environment and the relation to the observer (view, gaze), which allows for application in a number of fields. The recovered illumination map can be further applied to various computational tasks. This paper provides a comprehensive introduction on corneal imaging, and aims to show the potential of the topic and encourage advancement. It makes a number of contributions, including (1) a combined view on previously unrelated fields, (2) an overview of recent developments, (3) a detailed explanation on anatomic structures, geometric eye and corneal reflection modeling including multiple eye images, (4) a summary of our work and contributions to the field, and (5) a discussion of implications and promising future directions. The idea behind this paper is a geometric framework to solve persisting technical problems and enable non-intrusive interfaces and smart sensors for traditional, ubiquitous and ambient environments.