  We use tools from generalized complex geometry to develop the theory of SKT
(a.k.a. pluriclosed Hermitian) manifolds and more generally manifolds with
special holonomy with respect to a metric connection with closed skew-symmetric
torsion. We develop Hodge theory on such manifolds showing how the reduction of
the holonomy group causes a decomposition of the twisted cohomology. For SKT
manifolds this decomposition is accompanied by an identity between different
Laplacian operators and forces the collapse of a spectral sequence at the first
page. Further we study the deformation theory of SKT structures, identifying
the space where the obstructions live. We illustrate our theory with examples
based on Calabi--Eckmann manifolds, instantons, Hopf surfaces and Lie groups.
