Phenomena involving stable geometries of continuously varying objects are related topology. Therefore, topology finds a lot of applications in metamaterials in the target of manipulation of various scalar and vectorial waves. We focus on how topology induces unique phenomena in photonic and phononic metamaterials. These fields are on the rise and has made important developments in the past few years. Topological photonic and phononic systems are out of equilibrium systems which distinguishes them from the equilibrium topological electronic systems. This enables a series of new aspects such as non-Hermitian effects, nonlinear effects, local pump-probe, and non-Euclidean geometries. The study of topological metamaterials also finds its applications in topological lasers, waveguides, and sensors.