We consider the variety of Filippov (n-Lie) algebra structures on an (n + 1)-dimensional vector space. The group GL_n(K) acts on it, and we study the orbit closures with respect to the Zariski topology. This leads to the definition of Filippov algebra degenerations. We present some fundamental results on such degenerations, including trace invariants and necessary degeneration criteria. Finally, we classify all orbit closures in the variety of complex (n + 1)-dimensional Filippov n-ary algebras.