  Let X be a smooth projective variety over C. We find the natural notion of
semistable orthogonal bundle and construct the moduli space, which we
compactify by considering also orthogonal sheaves, i.e. pairs (E,\phi), where E
is a torsion free sheaf on X and \phi is a symmetric nondegenerate (in the open
set where E is locally free) bilinear form on E. We also consider special
orthogonal sheaves, by adding a trivialization \psi of the determinant of E
such that det(\phi)=\psi^2 ; and symplectic sheaves, by considering a form
which is skewsymmetric. More generally, we consider semistable tensors, i.e.
multilinear forms on a torsion free sheaf, and construct their projective
moduli space using GIT.
