In this talk I will sketch how techniques from descriptive set theory can be used to prove results about Lipschitz-free (hereafter free) spaces, such as

(1) if a separable Banach space $X$ contains a linearly isomorphic copy of every free space over a countably infinite complete discrete metric space, then $X$ contains a linearly isomorphic copy of every separable Banach space; and

(2) there exists a countably infinite complete discrete metric space whose free space fails the bounded approximation property (BAP).

The first result extends a recent theorem of Basset, Lancien and Proch\'azka. To the best of my knowledge, the second result yields the first free space having the approximation property but not the BAP, and having the Radon-Nikod\'ym property without being isomorphic to a dual space. Interestingly, I have no concrete description of the space; it is obtained indirectly via a `complexity argument'.