  This paper is devoted to the first systematic investigation of manifolds that
are Einstein for a connection with skew symmetric torsion. We derive the
Einstein equation from a variational principle and prove that, for parallel
torsion, any Einstein manifold with skew torsion has constant scalar curvature;
and if it is complete of positive scalar curvature, it is necessarily compact
and it has finite first fundamental group. The longest part of the paper is
devoted to the systematic construction of large families of examples. We
discuss when a Riemannian Einstein manifold can be Einstein with skew torsion.
We give examples of almost Hermitian, almost metric contact, and G2 manifolds
that are Einstein with skew torsion. For example, we prove that any
Einstein-Sasaki manifold and any 7-dimensional 3-Sasakian manifolds admit
deformations into an Einstein metric with parallel skew torsion.
