Interphases between solids are crucial for understanding the behavior of composite artificial and natural structures. Due to the thinness of these interphases, numerical modeling using finite element analysis may be challenging. Asymptotic techniques, modeling the interphase as an interface, are viable alternatives. In this paper, we review recent researches on imperfect interface laws modeling the multiphysic and damaging behavior of thin adhesive layers joining two adherents. Two main cases are considered: a linear multiphysics framework with a soft limit interface model, and the behavior of damaged structural elastic adhesives. The latter, considered as a particular case of the multiphysic setting, combines asymptotic theory and micromechanical homogenization to describe ductile and brittle damage in adhesives. A novel example of a composite block with an adhesive characterized by elastic soft damaging behavior under combined shear and tensile loads is provided.