After briefly reviewing the notion of geometries characterised by SU(3) structures, I will discuss a particular class known (in String Theory) as non-Kahler. I will explain how this arises both in Type II and Heterotic Supergravities. Two explicit constructions of these geometries will be discussed. One is a one-parameter solution corresponding to fivebranes wrapped on the two-sphere of the resolved conifold, that can be thought of as a non-Kahler analog of the conifold. The other is a general construction of one-parameter non-Kahler deformations of Calabi-Yau manifolds with a U(1) isometry, where the non-Abelian Yang-Mills field of the Heterotic is non-trivial. The presentation will be loosely based on the two papers: arXiv:0906.0591 and arXiv:1010.4031.