With the surging demand for ultra-reliable, low-latency, and ubiquitous connectivity in Sixth-Generation (6G) networks, Non-Terrestrial Networks (NTNs) emerge as a key complement to terrestrial networks by offering flexible access and global coverage. Despite the significant potential, NTNs still face critical challenges, including dynamic propagation environments, energy constraints, and dense interference. As a key 6G technology, Fluid Antennas (FAs) can reshape wireless channels by reconfiguring radiating elements within a limited space, such as their positions and rotations, to provide higher channel diversity and multiplexing gains. Compared to fixed-position antennas, FAs can present a promising integration path for NTNs to mitigate dynamic channel fading and optimize resource allocation. This paper provides a comprehensive review of FA-assisted NTNs. We begin with a brief overview of the classical structure and limitations of existing NTNs, the fundamentals and advantages of FAs, and the basic principles of FA-assisted NTNs. We then investigate the joint optimization solutions, detailing the adjustments of FA configurations, NTN platform motion modes, and resource allocations. We also discuss the combination with other emerging technologies and explore FA-assisted NTNs as a novel network architecture for intelligent function integrations. Furthermore, we delve into the physical layer security and covert communication in FA-assisted NTNs. Finally, we highlight the potential future directions to empower broader applications of FA-assisted NTNs.