With the rapid development of communications and computing, the concept of connected vehicles emerges to improve driving safety, traffic efficiency and infotainment experience. Due to the limited capabilities of sensors and information processing on a single vehicle, vehicular networks (VNETs) play a vital role for the realization of connected vehicles. To achieve the closer cooperation on the road, ultra-reliable and low-latency communications (URLLCs) become the indispensable components of future VNETs. However, traditional wireless networks are designed with the objective of maximizing spectral and energy efficiency regardless of adequate consideration of URLLCs. To this end, this article investigates the challenges and solutions of conceiving URLLC VNETs dedicated for connected vehicles. Specifically, the underlying application scenarios are first presented with various vehicular communication patterns. Then, the URLLC requirements and potential challenges are discussed for VNETs. We also discuss the solutions to establish URLLC VNETs from the aspects of physical and media access control layers, including the optimization frameworks for latency and reliability, as well as the case studies of employing such frameworks. Finally, several open research directions are pointed out.