The renaissance of deep learning has provided promising solutions to various tasks. While conventional deep learning models are constructed for a single specific task, multi-task deep learning (MTDL) that is capable to simultaneously accomplish at least two tasks has attracted research attention. MTDL is a joint learning paradigm that harnesses the inherent correlation of multiple related tasks to achieve reciprocal benefits in improving performance, enhancing generalizability, and reducing the overall computational cost. This review focuses on the advanced applications of MTDL for medical image computing and analysis. We first summarize four popular MTDL network architectures (i.e., cascaded, parallel, interacted, and hybrid). Then, we review the representative MTDL-based networks for eight application areas, including the brain, eye, chest, cardiac, abdomen, musculoskeletal, pathology, and other human body regions. While MTDL-based medical image processing has been flourishing and demonstrating outstanding performance in many tasks, in the meanwhile, there are performance gaps in some tasks, and accordingly we perceive the open challenges and the perspective trends. For instance, in the 2018 Ischemic Stroke Lesion Segmentation challenge, the reported top dice score of 0.51 and top recall of 0.55 achieved by the cascaded MTDL model indicate further research efforts in high demand to escalate the performance of current models.