Achieving fully autonomous driving with enhanced safety and efficiency relies on vehicle-to-everything (V2X) cooperative perception (CP), which enables vehicles to share perception data, thereby enhancing situational awareness and overcoming the limitations of the sensing ability of individual vehicles. V2X CP plays a crucial role in extending the perception range, increasing detection accuracy, and supporting more robust decision-making and control in complex environments. This article provides a comprehensive survey of recent developments in V2X CP, introducing mathematical models that characterize the perception process under different collaboration strategies. Key techniques for enabling reliable perception sharing, such as agent selection, data alignment, and feature fusion, are examined in detail. In addition, major challenges are discussed, including differences in agents and models, uncertainty in perception outputs, and the impact of communication constraints such as transmission delay and data loss. This article concludes by outlining promising research directions, including privacy-preserving artificial intelligence methods, collaborative intelligence, and integrated sensing frameworks to support future advancements in V2X CP.