The sixth-generation wireless networks are envisioned to deliver ubiquitous, seamless, and intelligent connectivity that reaches far beyond the limits of terrestrial infrastructure. Non-terrestrial networks (NTNs) are central to this vision, extending coverage to underserved regions, remote terrain, and disaster zones that terrestrial deployment cannot economically reach. However, NTN architecture faces numerous limitations: severe path loss over long distances, long propagation delays, large and time-varying Doppler shifts, limited visibility windows, and tight on-board energy and computing budgets. Semantic communication (SemCom), which conveys the meaning of data rather than its raw bit-level representation, is unusually well matched to these conditions: extreme compression rate for task-oriented eases bandwidth scarcity, deep joint source-channel coding prevents the cliff effect due to low signal-to-noise ratio, and generative-AI reconstructs content from sparse cues that survive rain-faded or blocked links. This observation, that each NTN limitation maps onto a SemCom property that addresses it, motivates our survey. We first walk through the NTN limitations one by one, pairing each with the SemCom design choices that complement it, then we organize the literature along three axes: the NTN platform, the semantic methodology, and the supporting techniques, and follow this with platform-by-platform deep dives on satellite-centric, UAV/HAPS-centric, and integrated SAGIN systems. The survey concludes by identifying open research problems, gaps in existing standards, and future directions, including the application of foundation models, energy-aware scheduling, and quantum-assisted SemCom for deep space communication.