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Vehicles are becoming more intelligent and automated. To achieve higher automation levels, vehicles are being equipped with more and more sensors. High data rate connectivity seems critical to allow vehicles and road infrastructure exchanging all these sensor data to enlarge their sensing range and make better safety related decisions. Connectivity also enables other applications such as infotainment or high levels of traffic coordination. Current solutions for vehicular communications though do not support the gigabit-per-second data rates. This presentation makes the case that millimeter wave communication is the only viable approach for high bandwidth connected vehicles. The motivation and challenges associated with using mmWave for vehicle-to-vehicle and vehicle-to-infrastructure applications are highlighted. Examples from recent work are provided including new theoretical results that enable mmWave communication in high mobility scenarios and innovative architectural concepts like position and radar-aided communication.