  We investigate the deformation theory of a class of generalized calibrations
in Riemannian manifolds for which the tangent bundle has reduced structure
group U(n), SU(n), G_2 and Spin(7). For this we use the property of the
associated calibration form to be parallel with respect to a metric connection
which may have non-vanishing torsion. In all these cases, we find that if there
is a moduli space, then it is finite dimensional.
  We present various examples of generalized calibrations that include almost
hermitian manifolds with structure group U(n) or SU(n), nearly parallel G_2
manifolds and group manifolds. We find that some Hopf fibrations are
deformation families of generalized calibrations. In addition, we give
sufficient conditions for a hermitian manifold (M,g,J) to admit Chern and
Bismut connections with holonomy contained in SU(n). In particular we show that
any connected sum of $k \geq 3$ copies of $S^3 \times S^3$ admits a hermitian
structure for which the restricted holonomy of a Bismut connection is contained
in SU(3).
