In this talk we present a new class of phases of matter called higher-order topological phases. Such systems exhibit gapped boundaries that are themselves lower-dimensional topological phases. Furthermore, they manifest topologically protected states at intersections between multiple surfaces. These states can, e.g., carry fractional charge, or be chiral propagating modes. We show that some of the simplest examples of these phases are connected to materials with quantized electric quadrupole moments, and illustrate how they can be realized in a variety of meta-material systems. We will also review the first experimental evidence for these phases in meta-material contexts. To characterize these new insulating phases of matter, we introduce a new type of invariant that had been previously overlooked, and we show how these invariants can predict a collection of new phenomena.
