Abstract: Consider the variety of pre-Lie algebra structures on a given n-dimensional vector space. The group GL_n(K) acts on it, so one can study the closure of the orbits with respect to the Zariski topology. This leads to the definition of pre-Lie algebra degenerations. Fundamental results on such degenerations, including invariants and necessary degeneration criteria will be given. Finally, as an example, I will sketch the proof of the classification of all orbit closures in the variety of complex 2-dimensional pre-Lie algebras.