Unlimited and seamless coverage as well as ultra-reliable and low-latency communications are vital for connected vehicles, in particular for new use cases like autonomous driving and vehicle platooning. In this paper, we propose a novel Space-Air-Ground integrated vehicular network (SAGiven) architecture to gracefully integrate the multi-dimensional and multi-scale context-information and network resources from satellites, High-Altitude Platform stations (HAPs), low-altitude Unmanned Aerial Vehicles (UAVs), and terrestrial cellular communication systems. One of the key features of the SAGiven is the reconfigurability of heterogeneous network functions as well as network resources. We first give a comprehensive review of the key challenges of this new architecture and then provide some up-to-date solutions on those challenges. Specifically, the solutions will cover the following topics: (1) space-air-ground integrated network reconfiguration under dynamic space resources constraints; (2) multi-dimensional sensing and efficient integration of multi-dimensional context information; (3) real-time, reliable, and secure communications among vehicles and between vehicles and the SAGiven platform; and (4) a holistic integration and demonstration of the SAGiven. Finally, it is concluded that the SAGiven can play a key role in future autonomous driving and Internet-of-Vehicles applications.