The proliferation of orbital sensing data and the real-time demands of emerging global services are pushing low Earth orbit (LEO) satellite networks from bent-pipe relays toward intelligent onboard processing. This shift confronts a fundamental tension, where modern artificial intelligence (AI) training paradigms require intensive computation, abundant memory, and stable connectivity, all of which are scarce under the dynamic topologies, intermittent links, and severe size, weight, and power (SWaP) constraints of satellite networks. Addressing this tension demands rethinking AI workflows, from training and inference to the emerging deployment of large language models (LLMs), under a constraint-aware design philosophy. To the best of our knowledge, this paper provides the first survey that jointly covers all three, whereas prior surveys examine these aspects in isolation. We begin by identifying the deployment challenge of satellite intelligence (SI) and deriving design principles for SI. We then examine three advancing paradigms: i) distributed training, where federated learning (FL) and split learning (SL) enable satellite-ground co-training under resource limitations; ii) collaborative inference, which exploits single-satellite, multi-satellite, and satellite-ground architectures to meet real-time service demands; and iii) on-orbit LLM adaptation and serving, which tackles the frontier challenge of deploying LLMs under severe orbital constraints and resource-aware serving. Finally, we identify open challenges and future directions. By bridging the AI and satellite networks, this paper provides a research roadmap for the implementation of space-native AI technologies.

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