A Positive Engagement, Continuously Variable Transmission (PECVT) allows for a continuously variable transmission ratio over a given range using positively engaged members, such as gear teeth, to transmit torque. This research is an investigation of PECVTs to establish a classification system and governing principles that must be satisfied for an embodiment to overcome the non-integer tooth problem. Results of an external patent search are given as examples of different concepts and PECVT embodiments that have been employed to negate the effects of the non-integer tooth problem. To classify all published and unpublished PECVT embodiments, a classification system is developed, based on how particular PECVT embodiments overcome the non-integer tooth problem. Two classes of PECVTs are defined: 1) the problem correction class and 2) the alternate device class. General principles that must be satisfied for a promising PECVT embodiment to exist in each class of PECVTs are also developed. These principles, along with the classification system, are the major contribution of this research. The principles describe what an embodiment in each of the PECVT classes must accomplish to negate the effects of the non-integer tooth problem. A product development phase integrated with TRIZ methodology is implemented to generate several concepts that satisfy the newly developed general principles and the product specifications that were also created. A screening and scoring process is used to eliminate less promising concepts and to find the most viable PECVT embodiment. An embodiment that only operates at preferred transmission ratios, where no meshing problems exist, proves to be the most promising concept based on the results of this methodology. The embodiment also utilizes cams and a differential device to provide the needed correction to the orientation of the driving members when misalignment occurs. This misalignment only occurs while transitioning between preferred operating ratios. A case study of the final embodiment developed by Vernier Moon Technologies and Brigham Young University is presented and analyzed to show how the final concepts ensure proper engagement without the effects of the non-integer tooth problem. The final embodiment is not the optimal solution but represents a conceptual design of an embodiment that satisfies the governing principles. The classification system and the governing principles that have been established are valid for all PECVT embodiments and will be valuable in future research. Future work yet to be conducted for this research, including an involutometry analysis, is discussed as well as other recommendations.