We study the algebraic dimension a(X) of a compact hyperk ¨ahler manifold of dimension 2n. We show that a(X) is at most
n unless X is projective. If a compact K¨ahler manifold with algebraic dimension 0 and Kodaira dimension 0 has a minimal model,
then only the values 0, n and 2n are possible. In case of middle dimension, the algebraic reduction is holomorphic Lagrangian. If
n = 2, then - without any assumptions - the algebraic dimension only takes the values 0, 2 and 4. The paper also gives structure results
for ”generalised hyperk¨ahler” manifolds and studies nef lines bundles.