  We give several construction methods and use them to produce many examples of
proper biharmonic maps including biharmonic tori of any dimension in Euclidean
spheres (Theorem 2.2, Corollaries 2.3, 2.4, and 2.6), biharmonic maps between
spheres (Theorem 2.9) and into spheres (Theorem 2.10) via orthogonal
multiplications and eigenmaps. We also study biharmonic graphs of maps, derive
the equation for a function whose graph is a biharmonic hypersurface in a
Euclidean space, and give an equivalent formulation of Chen's conjecture on
biharmonic hypersurfaces by using the biahrmonic graph equation (Theorem 4.1)
which paves a way for analytic study of the conjecture.
