Non-terrestrial network (NTN) is a trending topic in the field of communication, as it shows promise for scenarios in which terrestrial infrastructure is unavailable. Unmanned autonomous intelligent systems (UAISs), as a physical form of artificial intelligence (AI), have gained significant attention from academia and industry. These systems have various applications in autonomous driving, logistics, area surveillance, and medical services. With the rapid evolution of information and communication technology (ICT), 5G and beyond-5G communication have enabled numerous intelligent applications through the comprehensive utilization of advanced NTN communication technology and artificial intelligence. To meet the demands of complex tasks in remote or communication-challenged areas, there is an urgent need for reliable, ultra-low latency communication networks to enable unmanned autonomous intelligent systems for applications such as localization, navigation, perception, decision-making, and motion planning. However, in remote areas, reliable communication coverage is not available, which poses a significant challenge for intelligent systems applications. The rapid development of non-terrestrial networks (NTNs) communication has shed new light on intelligent applications that require ubiquitous network connections in space, air, ground, and sea. However, challenges arise when using NTN technology in unmanned autonomous intelligent systems. Our research examines the advancements and obstacles in academic research and industry applications of NTN technology concerning UAIS, which is supported by unmanned aerial vehicles (UAV) and other low-altitude platforms. Nevertheless, edge computing and cloud computing are crucial for unmanned autonomous intelligent systems, which also necessitate distributed computation architectures for computationally intensive tasks and massive data offloading. This paper presents a comprehensive analysis of the opportunities and challenges of unmanned autonomous intelligent systems in UAV NTN, along with NTN-based unmanned autonomous intelligent systems and their applications. A field trial case study is presented to demonstrate the application of NTN in UAIS.