The theory of Lipschitz extensions goes back to the 1930’s and the work of McShane, Whitney, and Kirrszbraun. In its simplest form, the theory states that if one has a real-valued Lipschitz function defined on a subset of Euclidean space, then this function can be extended to the entire space while preserving the Lipschitz constant. The extension, generally speaking, is not unique, and in the 1960’s Aronsson proposed searching for the locally best Lipschitz extension, also known as the absolutely minimal Lipschitz extension (AMLE). Since their introduction by Aronsson, AMLEs have been the subject of extensive research, both for their intrinsic interest and for their relationship to PDEs. Extending the notion of an AMLE to non-scalar valued functions, though, has proven to be difficult, with only a couple recent results making progress. In this talk, we will begin by exploring some of the history of Lipschitz extensions and AMLEs, making our way up to the present day. Then I will present a new result, obtained jointly with Erwan Le Gruyer, in which we seek to add to the progress on non-scalar valued AMLEs. In this result we prove a general theorem of existence of what we call a quasi-AMLE, which essentially means a function that behaves almost like an AMLE. What we lose in precision, we make up for in generality, as the theorem holds for several different classes of functions that have the Lipschitz extension property.