A Digital Twin is a digital replica of a living or non-living physical entity, and this emerging technology has attracted extensive attention from different industries during the past decade. Although a few Digital Twin studies have been conducted in the transportation domain very recently, there is no systematic research with a holistic framework connecting various mobility entities together. In this study, by leveraging both connected vehicle technology and cloud computing, an Mobility Digital Twin (MDT) framework is developed, which consists of three building blocks in the physical space (namely Human, Vehicle, and Traffic), and their associated Digital Twins in the digital space. The cloud architecture is built with Amazon Web Services (AWS) to accommodate the proposed MDT framework and to implement its digital functionalities of storage, modeling, learning, simulation, and prediction. The effectiveness of the MDT framework is shown through the case studies of three digital building blocks with their key microservices: the Human Digital Twin with user management and driver type classification, the Vehicle Digital Twin with cloud-based Advanced Driver-Assistance Systems (ADAS), and the Traffic Digital Twin with traffic flow monitoring and variable speed limit.