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Hyper-Kähler manifolds occupy a central place in the classification of algebraic varieties. They are even-dimensional and have trivial canonical bundle. In dimension two, there is—up to deformation—only one such example: the K3 surface. In higher dimensions, however, the picture becomes significantly more mysterious. In this talk, I report on work in progress towards bounding and ultimately classifying four-dimensional hyper-Kähler manifolds.