  We study the algebraic dimension a(X) of a compact hyperkaehler manfold of
dimension 2n. We show that a(X) is at most n unless X is projective. If a
compact Kaehler 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 gives structure results for "generalised hyperkaehler"
manifolds and studies nef lines bundles.
