Hypergraph neural networks (HGNNs) have emerged as a powerful framework for modeling complex, high-order relationships in data that go beyond traditional graph structures. This review examines the implementation and applications of HGNNs across diverse domains. We analyze how HGNNs have been adapted to capture multi-way interactions in fields such as social network analysis, bioinformatics, computer vision, and recommendation systems. Key architectural variations and training approaches are discussed, along with domain-specific challenges and solutions. We also explore how HGNNs compare to traditional graph neural networks in terms of expressiveness and computational efficiency. Finally, we highlight open research questions and promising future directions for expanding the use of HGNNs to new problem domains. This comprehensive overview aims to provide researchers and practitioners with insights into effectively leveraging hypergraph-based deep learning for complex relational data.