Recent significant advances in self-interference cancellation techniques pave the way for the deployment of full-duplex wireless transceivers capable of concurrent transmission and reception on the same channel. Despite the promise to theoretically double the spectrum efficiency, full-duplex prototyping in off-the-shelf chips of mobile devices is still in its infancy, mainly because of the challenges in mitigating self-interference to a tolerable level and the strict hardware constraints. In this article, we argue in favor of embedding full-duplex radios in onboard units of future vehicles. Unlike the majority of mobile devices, vehicular onboard units are good candidates to host complex full-duplex transceivers because of their virtually unlimited power supply and processing capacity. Taking into account the effect of imperfect self-interference cancellation, we investigate the design implications of full-duplex devices at the higher-layer protocols of next-generation vehicular networks and highlight the benefits they could bring with respect to half-duplex devices in some representative use cases. Early results are also provided that give insight into the impact of self-interference cancellation on vehicle-to-roadside communications, and showcase the benefits of FD-enhanced medium access control protocols for vehicle-to-vehicle communications supporting crucial road safety applications.