Abstract: As a generalization of traditional graph structures, hypergraphs provide a powerful mathematical framework for modeling complex, high-order, and many-to-many relationships among entities. With the rapid advancement of artificial intelligence (AI) and machine learning (ML), research on hypergraph algorithms has gained increasing momentum, spanning both theoretical foundations and practical applications. Motivated by this trend, this survey offers a systematic and comprehensive overview of hypergraph algorithms, along with cloud computing scenarios in which hypergraph modeling can be effectively applied. The paper begins by introducing the fundamentals of hypergraphs and analyzing the structural distinctions between hypergraphs and traditional graphs, highlighting their practical implications for cloud computing tasks. We then review theoretical advances, with particular attention to partitioning, coloring, and isomorphism, followed by a comprehensive overview of hypergraph learning methods, including spectral methods and neural network based techniques. In addition, we analyze the types of scenarios in cloud computing where hypergraph methods can be effectively applied, with emphasis on data center networking, traffic prediction, resource scheduling, data management, anomaly detection, and cloud security. With the advancement of AI-driven learning methods, hypergraph-based models are increasingly capable of capturing high-order dependencies, enabling more timely and accurate decision making in complex cloud environments. Last but not least, we outline the major challenges and opportunities for future research. This paper provides critical insights into the role of hypergraphs in addressing complex computational and organizational challenges across multidisciplinary fields.