Advances in materials science and engineering through bio-inspiration, at both the micro- and nanoscales, have flourished over recent years. By understanding principles used in nature to produce adhesives and other substances of interest, the field of bio-inspired engineering has emerged as an important area of innovation. In this review, we will focus on bio-adhesives based on three main mechanisms of generating attachment: dry, wet, and chemical adhesion. Dry adhesion, involving micro- to nanoscale filamentous structures, is used by many insects and reptiles to rapidly climb surfaces. Tree frogs and some insects make use of wet adhesion by leveraging capillary forces through the design of attaching structures that increases liquid drainage, and hence increases frictional contact. Finally, chemical adhesion is used by many plants and mollusks, which secrete adhesives composed of proteins, polysaccharides and carbohydrates to generate the strong forces necessary for adhesion. This paper reviews recent discoveries in animal and plant bio-adhesives, and details the mechanisms used in several representative biological systems. We extend the review to include the fundamental principles functioning in each form of adhesion at the micro- and nanoscales. This fast emerging research area has significant implications in the future design of bio-inspired adhesives, and offers further potential for a variety of applications.