Near-space communication network (NS-ComNet), as an indispensable component of sixth-generation (6G) and beyond mobile communication systems and the space-air-ground-sea integrated network (SAGSIN), demonstrates unique advantages in wide-area coverage, long-endurance high-altitude operation, and highly flexible deployment. This paper presents a comprehensive review of NS-ComNet for 6G and beyond era. Specifically, by contrasting satellite, low-altitude unmanned-aerial-vehicle (UAV), and terrestrial communications, we first elucidate the background and motivation for integrating NS-ComNet into 6G network architectures. Subsequently, we review the developmental status of near-space platforms, including high-altitude balloons, solar-powered UAVs, and stratospheric airships, and analyze critical challenges faced by NS-ComNet. To address these challenges, the research focuses on key enabling technologies such as topology design, resource and handover management, multi-objective joint optimization, etc., with particular emphasis on artificial intelligence techniques for NS-ComNet. Finally, envisioning future intelligent collaborative networks that integrate NS-ComNet with satellite-UAV-terrestrial systems, we explore promising directions. This paper aims to provide technical insights and research foundations for the systematic construction of NS-ComNet and its deep deployment in the 6G and beyond era.