Connected and autonomous vehicles (CAVs) can reduce human errors in traffic accidents, increase road efficiency, and execute various tasks ranging from delivery to smart city surveillance. Reaping these benefits requires CAVs to autonomously navigate to target destinations. To this end, each CAV's navigation controller must leverage the information collected by sensors and wireless systems for decision-making on longitudinal and lateral movements. However, enabling autonomous navigation for CAVs requires a convergent integration of communication, control, and learning systems. The goal of this article is to explicitly expose the challenges related to this convergence and propose solutions to address them in two major use cases: Uncoordinated and coordinated CAVs. In particular, challenges related to the navigation of uncoordinated CAVs include stable path tracking, robust control against cyber-physical attacks, and adaptive navigation controller design. Meanwhile, when multiple CAVs coordinate their movements during navigation, fundamental problems such as stable formation, fast collaborative learning, and distributed intrusion detection are analyzed. For both cases, solutions using the convergence of communication theory, control theory, and machine learning are proposed to enable effective and secure CAV navigation. Preliminary simulation results are provided to show the merits of proposed solutions.