We study Riemannian manifolds admitting parallel spinors with respect to a metric connection with totally skew-symmetric torsion. We consider the cases when the holonomy group is contained in G={SU(n),Sp(n),U(n)xId, G_2, Spin(7)}. We find necessary and sufficient conditions of the existence of a G-connection with torsion 3-form. We show that it is unique and study the underlying geometry. We obtain a generalization of Calabi-Yau manifolds - Calabi-Yau manifolds with torsion. We show the existence of a SU(n)-connection with torsion 3-form on some compact complex (non-Kaehler) manifold with zero first Chern class and conjecture that this phenomena holds in general.