What is...?

CAT(0) cube complexes are cell complexes whose cells are cubes, whose naturally defined metric is non-positively curved in some precise sense. 

They can be equivalently defined in a variety of ways, which a priori looks very different. They naturally arise, as e.g. configuration spaces of discrete systems. They also played an important role in resolution of some long-standing open problems in topology of 3-dimensional manifolds.