Tropical cyclones (TCs) are highly destructive weather systems, typically accompanied by heavy rainfall, extreme winds and storm surges, significantly impacting residents' safety and property, and economic development. Accurate and efficient TC monitoring methods are of great importance for disaster prevention and mitigation, and emergency response. In recent years, multi-source remote sensing has provided important data sources for TC monitoring. In addition, the rapid development of artificial intelligence, particularly deep learning, has demonstrated tremendous potential in TC monitoring due to its powerful feature learning and representation capabilities. Therefore, this paper provides a comprehensive review of monitoring TCs using multi-source data and deep learning. This review first presents an overview of TC monitoring tasks, including TC centre identification, intensity estimation, and wind radii estimation. Subsequently, the widely used multi-source data types and a list of publicly available datasets are provided. Towards typical TC monitoring tasks, this review demonstrates how to transform these tasks into fundamental tasks. The corresponding network architectures, specific models, and multi-source data involved, and detailed implementations are summarised. In addition, this review provides insights into challenges and future directions, focusing on aspects including multi-task learning, multi-modal data fusion, incomplete data modalities, lightweight models, and physics-informed models for TC monitoring.