Leveraging connectivity for controlling connected automated vehicles (CAVs) has great potential for improving the safety and efficiency of transportation. In this paper, we study the safety of connected cruise control (CCC), wherein CAVs respond to multiple preceding vehicles via vehicle-to-everything (V2X) connectivity. Using control barrier function theory, we analyze how connectivity to vehicles farther ahead can be leveraged to improve the CAV's safety, and we propose safety-critical CCC by minimally modifying efficient but not always safe CCC designs. We use simulations to evaluate the proposed safety-critical CCC with respect to safety, energy efficiency and string stability. We also study mixed traffic, and show that increasing the penetration of CAVs can significantly improve safety and performance of road transportation systems. © 2024 IEEE.