Any spacetime containing an extremal Killing horizon has an associated near-horizon geometry. This is itself a spacetime that encodes key properties of the horizon, such as its conserved charges, and is an important tool for studying the classification of extremal black holes. In this talk, we discuss an analogue of Hawking’s rigidity theorem for extremal horizons, valid in arbitrary dimension and for general matter obeying the null energy condition. We then turn to the classification problem in five-dimensional Einstein-Maxwell theory and present new charged and rotating near-horizon geometries in closed form, based on joint work with Jun Liu. Finally, we propose a definition of higher multiplicity horizons and discuss its implications for the corresponding Aretakis instability.