Fifteen years since its inception, the Kitaev model still boasts only a narrow group of material realizations. The progress in studying and understanding one of them, lithium iridate available in three polymorphs that host strong Kitaev interactions on spin lattices of different dimensionality and topology, is reviewed. Feasibility, effectiveness, and repercussions of tuning strategies based on the application of external pressure and chemical substitutions are also discussed.