This paper is a survey of finiteness results in hyperkahler geometry. We review some classical theorems by Sullivan, Kollar-Matsusaka, Huybrechts, as well as theorems in the recent literature by Charles, Sawon, and joint results of the author with Verbitsky. We also strengthen a finiteness theorem of the author. These are extended notes of the authors talk during the closing conference of the Simons Semester in the Banach Center in Bc{e}dlewo, Poland.
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