The stability of a composite interface of roughness-induced superhydrophobic surfaces is studied. To have high contact angle and low contact angle hysteresis, superhydrophobic surfaces should be able to form a composite interface with air pockets in the valleys between asperities (pillars). However, the composite interface may be unstable and can be irreversibly transformed into a homogeneous interface. We formulate a stability criterion and analyze the stability of the composite interface for several typical roughness profiles. To resist destabilizing mechanisms, multiscale (hierarchical) roughness is required. Such multiscale roughness is found in natural and artificial superhydrophobic surfaces.