Although Deep neural networks (DNNs) have shown a strong capacity to solve large-scale problems in many areas, such DNNs are hard to be deployed in real-world systems due to their voluminous parameters. To tackle this issue, Teacher-Student architectures were proposed, where simple student networks with a few parameters can achieve comparable performance to deep teacher networks with many parameters. Recently, Teacher-Student architectures have been effectively and widely embraced on various knowledge distillation (KD) objectives, including knowledge compression, knowledge expansion, knowledge adaptation, and knowledge enhancement. With the help of Teacher-Student architectures, current studies are able to achieve multiple distillation objectives through lightweight and generalized student networks. Different from existing KD surveys that primarily focus on knowledge compression, this survey first explores Teacher-Student architectures across multiple distillation objectives. This survey presents an introduction to various knowledge representations and their corresponding optimization objectives. Additionally, we provide a systematic overview of Teacher-Student architectures with representative learning algorithms and effective distillation schemes. This survey also summarizes recent applications of Teacher-Student architectures across multiple purposes, including classification, recognition, generation, ranking, and regression. Lastly, potential research directions in KD are investigated, focusing on architecture design, knowledge quality, and theoretical studies of regression-based learning, respectively. Through this comprehensive survey, industry practitioners and the academic community can gain valuable insights and guidelines for effectively designing, learning, and applying Teacher-Student architectures on various distillation objectives.