<p>Multi-tier aerial communication networks (MACNs), integrating satellites, high-altitude platforms, and unmanned aerial vehicles, are emerging as a cornerstone of next-generation global connectivity. Their promise of resilient and ubiquitous coverage, however, is hindered by highly dynamic topologies, severe energy and computational constraints, environment-sensitive channels, diverse quality-of-service requirements, and limited real-world validation. Artificial intelligence (AI) has increasingly been explored as a flexible framework to address these challenges, enabling adaptive routing, distributed computing and task offloading, handover management, intelligent resource allocation, and large-scale network optimization. This survey provides a comprehensive and structured review of methods for MACNs, with particular emphasis on AI-driven solutions and their relationship to classical and hybrid alternatives. We critically evaluate representative approaches in terms of scalability, efficiency, data demands, and practical deployability, and identify emerging trends such as graph neural networks with reinforcement learning for dynamic routing, predictive learning for mobility management, and federated learning for distributed computation. Persistent challenges remain in lightweight edge intelligence, real-world testbeds, reproducible benchmarking, and simulation-to-deployment transfer. To address these issues, we offer a research roadmap emphasizing compressible and interpretable models, standardized benchmarks, realistic validation, and hybrid designs that balance adaptability with computational and energy overhead. Finally, we identify open challenges and future research directions, offering insights into the design of AI-driven MACNs that are efficient, scalable, and adaptive to evolving network and service demands.</p>