Hyperkähler metrics on K3 surfaces give rise to rational curves of degree 2 in the K3 period domain, so-called "twistor lines". While these are used in the proofs of many deep results, their existence also implies that the group of isometries of the K3 lattice does not act properly discontinuously on the period domain, preventing a moduli space of unpolarised complex K3 surfaces to exist. I will report on work in progress with Martin Schwald (Essen), in which we study the cycle space of the K3 period domain. This space parametrises twistor lines as part of its real locus, but also all their degenerations and complex deformations as submanifolds of the period domain. I will explain how many foundational problems regarding the moduli theory of K3s disappear when passing to the cycle space and also indicate how the original version of Penrose's Twistor Theory (the "nonlinear graviton" construction) can be used to understand what kind of geometric structure a small complex deformation of an honest twistor line corresponds to.