Quantum satellite communication (QSC) is emerging as a strategic technology for secure global networking and long-distance quantum connectivity. This review prioritizes the major challenges that still hinder large-scale deployment, including atmospheric loss, beam pointing and tracking, payload constraints, synchronization, scalability, and integration with terrestrial infrastructure. To contextualize these issues, we provide only a concise overview of the core concepts and enabling technologies behind QSC, together with representative milestones such as the Micius mission. Building on this background, the paper surveys recent advances in protocols, hybrid space--terrestrial architectures, turbulence mitigation, and AI-assisted optimization. It then examines future directions, including quantum Internet integration, daylight operation, satellite-supported repeaters, and space-based quantum computing. By centering the discussion on open technical bottlenecks and emerging research trajectories, this review aims to support researchers and engineers working toward practical and resilient QSC systems.