Special geometries, such as Calabi-Yau manifolds, play a central role in multiple areas of string theory, as well as gravitational theories more generally. The goal of these lectures is to introduce some of the formalism and tools useful for characterising such geometries, pitched at the level of a starting PhD student. We will start with purely geometrical backgrounds using the general notions of a G-structure and special holonomy and then will go on to describe backgrounds that also have non-trivial fluxes. We will be guided by applications to string phenomenology and the AdS/cft correspondence.