Adhesion to wet and dynamic surfaces, including biological tissues, is important in many fields but has proven to be extremely challenging. Existing adhesives are cytotoxic, adhere weakly to tissues, or cannot be used in wet environments. We report a bioinspired 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 energies on wet surfaces as compared with those of 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.