A GL-variety is a kind of infinite-dimensional variety on which the infinite general linear group GL acts. These varieties have surprising finiteness properties (e.g., Draisma proved that the descending chain condition holds for GL-stable closed subsets) and they can be useful for studying asymptotic properties in algebra (e.g., Stillman's conjecture). The basic theory of these varieties has been developed over the last few years. I will explain a somewhat deeper result, obtained recently by Bik, Draisma, Eggermont, and me, which constructs a sufficient supply of curves on GL-varieties. I will not assume any prior knowledge about GL-varieties.