Survey of finiteness results for hyperk\"ahler manifolds

This paper is a survey of finiteness results in hyperk"ahler geometry. We review some classical theorems by Sullivan, Koll'ar-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 author’s talk during the closing conference of the Simons Semester in the Banach Center in B\c{e}dlewo, Poland.