  We study biharmonic hypersurfaces and biharmonic submanifolds in a Riemannian
manifold. One of interesting problems in this direction is Chen's conjecture
which says that any biharmonic submanifold in a Euclidean space is minimal.
From the invariant equation for biharmonic submanifolds, we derive a
fundamental identity involving the mean curvature vector field, and using this,
we prove Chen's conjecture on biharmonic submanifolds in a Euclidean space.
More generally, it is proved that any biharmonic submanifold in a space form of
nonpositively sectional curvature is minimal. Furthermore we provide
affirmative partial answers to the generalized Chen's conjecture and
Balmu\c{s}-Montaldo-Oniciuc conjecture.
