I will describe a how to construct the supergravity solution for a pair of NS5-branes, intersecting on R^{1,3} and localised in all directions except a single transverse circle. Such solutions are rare, and yet may have many potential applications in gauge-gravity duality and model building. The construction invovles developing an explicit map between the conifold metric and the geometry of two intersecting NS5-branes. I will also describe additional intersecting brane solutions corresponding to smoothing out the intersection by, for example, separating branes or putting a metric on the intersection.