  Let X be a compact hyperk\"ahler manifold containing a complex torus L as a
Lagrangian subvariety. Beauville posed the question whether X admits a
Lagrangian fibration with fibre L. We show that this is indeed the case if X is
not projective. If X is projective we find an almost holomorphic Lagrangian
fibration with fibre L under additional assumptions on the pair (X, L), which
can be formulated in topological or deformation-theoretic terms. Moreover, we
show that for any such almost holomorphic Lagrangian fibration there exists a
smooth good minimal model, i.e., a hyperk\"ahler manifold birational to X on
which the fibration is holomorphic.
