Adhesion to wet and dynamic surfaces, including biological tissues, is important in many fields, but has proven extremely challenging. Existing adhesives are either cytotoxic, adhere weakly to tissues, or cannot be utilized in wet environments. We report a bio-inspired design for adhesives consisting of two layers: an adhesive surface and a dissipative matrix. The former adheres to the substrate by electrostatic interactions, covalent bonds, and physical interpenetration. The latter amplifies energy dissipation through hysteresis. The two layers synergistically lead to higher adhesion energy on wet surfaces than existing adhesives. Adhesion occurs within minutes, independent of blood exposure, and compatible with in vivo dynamic movements. This family of adhesives may be useful in many areas of application, including tissue adhesives, wound dressings and tissue repair.