In this talk we will discuss some aspects of Riemannian geometry where the chosen connection has a nonzero three-form as its torsion tensor. We will show how to decompose the curvature tensor for such a connection in four dimensions and mention how this motivates our definition of
Einstein manifolds with skew torsion. Relations with Einstein-Wyel and Hermitian geometry will also be presented. We will also consider the link with instanton moduli spaces and see how to a three form on the base manifold induces a three-form on the moduli space and give particular attention to the case where the base manifold is the four-sphere with the round metric.