Many organisms in nature thrive in intricate habitats through their unique bio-adhesive surfaces, facilitating tasks such as capturing prey and reproduction. It's important to note that the remarkable adhesion properties found in these natural biological surfaces primarily arise from their distinct micro- and nanostructures and/or chemical compositions. To create artificial surfaces with superior adhesion capabilities, researchers delve deeper into the underlying mechanisms of these captivating adhesion phenomena to draw inspiration. This article provides a systematic overview of various biological surfaces with different adhesion mechanisms, focusing on surface micro- and nanostructures and/or chemistry, offering design principles for their artificial counterparts. Here, the basic interactions and adhesion models of natural biological surfaces are introduced first. This will be followed by an exploration of research advancements in natural and artificial adhesive surfaces including both dry adhesive surfaces and wet/underwater adhesive surfaces, along with relevant adhesion characterization techniques. Special attention is paid to stimulus-responsive smart artificial adhesive surfaces with tunable adhesive properties. The goal is to spotlight recent advancements, identify common themes, and explore fundamental distinctions to pinpoint the present challenges and prospects in this field.