Many organisms have attachment organs with excellent functions, such as adhesion, clinging, and grasping, as a result of biological evolution to adapt to complex living environments. From nanoscale to macroscale, each type of adhesive organ has its own underlying mechanisms. Many biological adhesive mechanisms have been studied and can be incorporated into robot designs. This paper presents a systematic review of reversible biological adhesive methods and the bioinspired attachment devices that can be used in robotics. The study discussed how biological adhesive methods, such as dry adhesion, wet adhesion, mechanical adhesion, and sub-ambient pressure adhesion, progress in research. The morphology of typical adhesive organs, as well as the corresponding attachment models, is highlighted. The current state of bioinspired attachment device design and fabrication is discussed. Then, the design principles of attachment devices are summarized in this article. The following section provides a systematic overview of climbing robots with bioinspired attachment devices. Finally, the current challenges and opportunities in bioinspired attachment research in robotics are discussed.