Topological phases of matter are low-temperature quantum phases that exhibit robust quantized transport properties and anomalous states at their boundaries. In this colloquium, I will describe the newly discovered family of "higher-order topological phases", characterized by anomalous boundary properties of at least two dimensions less than the dimension of their protecting bulk. I will describe their origins in terms of quantized higher multipole moments in crystalline insulators, their most salient properties in insulators, superconductors, and photonic systems, as well as their experimental demonstration in real and synthetic materials.