The interface is the separatrix between two media.It is only as an extreme in the model descriptions to be regarded as beingbilateral. There are many cases in which its another extreme, i.e., thesocalled unilateral model may possibly be more close to the reality.The propagation of elastic wave under unilateral interface are still not sufficiently investigated. It is the purpose ofthis paper to present a review on the current situations,problems and prospects in this research sphere.