Graph-based Retrieval-Augmented Generation (RAG) typically operates on binary Knowledge Graphs (KGs). However, decomposing complex facts into binary triples often leads to semantic fragmentation and longer reasoning paths, increasing the risk of retrieval drift and computational overhead. In contrast, n-ary hypergraphs preserve high-order relational integrity, enabling shallower and more semantically cohesive inference. To exploit this topology, we propose HyperRAG, a framework tailored for n-ary hypergraphs featuring two complementary retrieval paradigms: (i) HyperRetriever learns structural-semantic reasoning over n-ary facts to construct query-conditioned relational chains. It enables accurate factual tracking, adaptive high-order traversal, and interpretable multi-hop reasoning under context constraints. (ii) HyperMemory leverages the LLM's parametric memory to guide beam search, dynamically scoring n-ary facts and entities for query-aware path expansion. Extensive evaluations on WikiTopics (11 closed-domain datasets) and three open-domain QA benchmarks (HotpotQA, MuSiQue, and 2WikiMultiHopQA) validate HyperRAG's effectiveness. HyperRetriever achieves the highest answer accuracy overall, with average gains of 2.95% in MRR and 1.23% in Hits@10 over the strongest baseline. Qualitative analysis further shows that HyperRetriever bridges reasoning gaps through adaptive and interpretable n-ary chain construction, benefiting both open and closed-domain QA. Our codes are publicly available at https://github.com/Vincent-Lien/HyperRAG.git.