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We exhibit the first non-trivial concrete examples of Gromov-Hausdorffcompactifications of moduli spaces of K\"ahler-Einstein Fano manifolds in allcomplex dimensions bigger than two (Fano K-moduli spaces). We also discusspotential applications to explicit study of moduli spaces of K-stable Fanomanifolds with large anti-canonical volume. Our arguments are based on recentprogress about the geometry of metric tangent cones and on related ideas aboutthe algebro-geometric study of singularities of K-stable Fano varieties.

Explicit Gromov-Hausdorff compactifications of moduli spaces of K\"ahler-Einstein Fano manifolds