  One central problem in real algebraic geometry is to classify the real
structures of a given complex manifold. We address this problem for compact
hyperk\"ahler manifolds by showing that any such manifold admits only finitely
many real structures up to equivalence. We actually prove more generally that
there are only finitely many, up to conjugacy, faithful finite group actions by
holomorphic or anti-holomorphic automorphisms (the so-called Klein actions). In
other words, the automorphism group and the Klein automorphism group of a
compact hyperk\"ahler manifold contain only finitely many conjugacy classes of
finite subgroups. We furthermore answer a question of Oguiso by showing that
the automorphism group of a compact hyperk\"ahler manifold is finitely
presented.
