Amino acid-based poly(ester urea) (PEU) copolymers functionalized with pendant catechol groups that address the need for strongly adhesive yet degradable biomaterials have been developed. Lap-shear tests with aluminum adherends demonstrated that these polymers have lap-shear adhesion strengths near 1 MPa. An increase in lap-shear adhesive strength to 2.4 MPa was achieved upon the addition of an oxidative cross-linker. The adhesive strength on porcine skin adherends was comparable with commercial fibrin glue. Interfacial energies of the polymeric materials were investigated via contact angle measurements and Johnson–Kendall–Roberts (JKR) technique. The JKR work of adhesion was consistent with contact angle measurements. The chemical and physical properties of PEUs can be controlled using different diols and amino acids, making the polymers candidates for the development of biological glues for use in clinical applications.