Drug repurposing, the process of identifying new therapeutic uses for existing drugs, offers a promising strategy to accelerate drug development by significantly reducing costs, time, and risks compared to de novo drug discovery. The increasing availability of large-scale biomedical data has catalysed the development of computational approaches to systematically identify and prioritise repurposing candidates. This survey reviews the state-of-the-art computational methodologies, with a particular focus on network medicine and machine learning-based techniques. We discuss key approaches such as pathway-based analysis, network proximity, matrix factorisation, and the growing application of deep learning, particularly Graph Neural Networks (GNNs), which leverage complex biomedical networks. The paper explores how these methodsutilisee heterogeneous data—including drug-target interactions, gene-disease associations, and molecular structures—to generate repurposing hypotheses. Furthermore, we outline the primary challenges in the field, including data integration, model generalizability, and the need for explainability, and discuss future directions, such as the integration of multi-modal data and the development of more sophisticated, interpretable AI models.