Underwater adhesion has been the focus of many recent developments motivated by potential biomedical applications. Although most literature on underwater adhesives has focused on strong covalent chemistries, soft materials based on weak molecular interactions have gained interest. Instead of relying on potentially toxic chemical crosslinking reactions to form covalent bonds, these materials are often sticky due to their soft, viscoelastic nature, in a similar manner to soft hydrophobic Pressure-Sensitive Adhesives (PSAs). In this review, we critically discuss the state-of-the-art in the design and characterization of soft viscoelastic coacervates and gels based on specific weak molecular interactions for underwater adhesion. From the perspectives of materials science and mechanics, we investigate the relationships between the composition and structure of these materials and their underwater viscoelastic and adhesive properties. An originality of our review lies in the analogies and comparisons we draw with PSAs as well-understood hydrophobic self-adhesive counterparts of the relatively hydrophilic underwater adhesives discussed here. Considering current literature, a criterion has been proposed to distinguish hydrophilic and hydrophobic adhesives. The insights from this review are condensed into detailed guidelines for the design of future soft underwater adhesives. We conclude the review with important open questions and the perspectives of the field.