I will define permutation-equivariant K-theoretic Gromov-Witten invariants associated to a compact complex manifold X. They were recently introduced by Givental. Roughly speaking, by ``mirror formulae" I mean that certain hypergeoemetric series associated to X are generating series of these invariants. I will show one can write K-theoretic mirror formulae for various classes of manifolds (toric fibrations, hypersurfaces given by zero sections of convex line bundles). Time permitting, I will explain two ways of proving these results: using K-theoretic torus localization, where possible (done by Givental) or, more generally, twisted K-theoretic GW theory (myself).