Lipschitz-free spaces provide a linear framework for the study of metric spaces: to every pointed metric space $M$ one associates a Banach space $\mathcal{F}(M)$ whose dual is the space $\operatorname{Lip}_0(M)$ of real-valued Lipschitz functions vanishing at the distinguished point.

In this talk I will introduce Lipschitz-free spaces and discuss them from the perspective of Banach space geometry. I will focus on two open problems. First, given a metric space $M$ of density continuum, is $\operatorname{Lip}_0(M)$ weak$^*$-separable? This question is closely related to the structure theory of nonseparable Banach spaces. Second, is a Lipschitz-free space Schur if and only if it is quantitatively Schur? If so, what is the optimal quantitative constant? The talk is based on two recent papers: the first joint with L. Candido and B. Vejnar, and the second joint with O. Kalenda.