<p>This paper investigates interfacial wave propagation in two configurations: a semi-infinite metamaterial over an elastic half-space and a metaelastic layer atop an elastic half-space, considering both smooth and welded contact conditions. Auxetic and triple negative effective properties are considered for the metamaterial. Auxetic metamaterials are characterized by a negative Poisson’s ratio that gives rise to distinctive wave behaviors. The study also examines metamaterials with triple negative effective properties, where mass density, P-wave modulus, and shear modulus are simultaneously negative. The role of auxeticity on interfacial wave propagation in triple negative materials is also studied. The results offer a theoretical foundation for the design and development of elastic wave devices utilizing auxetic and triple negative metamaterials.</p>