This paper presents the foundation for a theory of presence that seeks to answer important questions about telepresence and virtual presence. The theory, which develops the definition of telepresence to include virtual presence as a special case, permits the precise definition of various types and degrees of telepresence. General categories of telepresence are defined, using different types of presence that are proposed in the paper. Three types of specifications are used to make the definitions more precise: (1) a set of tasks, (2) a transformation imposed on the human operator's control output and sensory input, and (3) a transformation of the region of presence. The proposed quantitative measure of telepresence involves both objective and subjective measures. The degree of (objective) telepresence is equal to the probability of successfully completing a specified task. The degree of subjective telepresence is equal to the probability that a human operator perceives that he or she is physically present in a given remote environment. The measure of subjective telepresence involves a psychophysical test and is analyzed using signal detection theory. Real-world complications are addressed and a practical example of a subjective telepresence test is described.