Accurate perception is essential for advancing autonomous driving and addressing safety challenges in modern transportation systems. Despite significant advancements in computer vision for object recognition, current perception methods still face difficulties in complex real-world traffic environments. Challenges such as physical occlusion and limited sensor field of view persist for individual vehicle systems. Cooperative Perception (CP) with Vehicle-to-Everything (V2X) technologies has emerged as a solution to overcome these obstacles and enhance driving automation systems. While some research has explored CP's fundamental architecture and critical components, there remains a lack of comprehensive summaries of the latest innovations, particularly in the context of V2X communication technologies. To address this gap, this paper provides a comprehensive overview of the evolution of CP technologies, spanning from early explorations to recent developments, including advancements in V2X communication technologies. Additionally, a contemporary generic framework is proposed to illustrate the V2X-based CP workflow, aiding in the structured understanding of CP system components. Furthermore, this paper categorizes prevailing V2X-based CP methodologies based on the critical issues they address. An extensive literature review is conducted within this taxonomy, evaluating existing datasets and simulators. Finally, open challenges and future directions in CP for autonomous driving are discussed by considering both perception and V2X communication advancements.