Artificial intelligence (AI) is expected to play a pivotal role in assisting human air traffic controllers, helping them manage the increasing demands of air traffic amid rising capacity challenges. By augmenting their capabilities in this safety-critical and human-intensive field, AI has the potential to bring significant transformation. This systematic review consolidates and analyzes existing research on human-AI interactions in air traffic management (ATM), with a focus on understanding the characteristics of human-AI hybrid (HAH) systems. It examines their role as enhancement tools across various operational levels and outlines their implementation through key phases, including conceptualization, development, evaluation, and training. Additionally, the review highlights current research gaps and provides recommendations for advancing HAH systems in ATM. By offering a comprehensive synthesis of the existing literature, this study lays a strong foundation for the future design and validation of HAH systems, contributing to enhanced safety and operational efficiency in air traffic management.