A conventional topological insulator (TI) has gapped bulk states but gapless edge states. The emergence of the gapless edge states is dictated by the bulk topological invariant of the insulator and the preservation of relevant symmetries. Over the past four years, a new type of TI has been found, which hosts gapless hinge or corner states, rather than edge states. These unconventional TIs, termed higher-order TIs (HOTIs), are common among crystalline and quasi-crystalline materials. Higher-order band topology expands our previous understanding of topological phases and provides unprecedented lower-dimensional boundary states for devices. Here, we review the principles, theories and experimental realizations of HOTIs for both electrons and classical waves. There is an emphasis on the development of HOTIs in photonic, phononic and circuit systems owing to their special contributions to these fields. From these discussions, we remark on trends and challenges in the field and the impact of higher-order band topology on other scientific disciplines.

传统的拓扑绝缘体 (TI) 具有带隙的体态，但具有无隙的边缘态。无间隙边缘状态的出现取决于绝缘体的体拓扑不变量和相关对称性的保持。在过去四年中，发现了一种新型 TI，它承载无间隙铰链或角状态，而不是边缘状态。这些非常规 TI，称为高阶 TI (HOTI)，在晶体和准晶体材料中很常见。高阶能带拓扑扩展了我们之前对拓扑相的理解，并为器件提供了前所未有的低维边界状态。在这里，我们回顾了电子和经典波的 HOTI 的原理、理论和实验实现。重点是光子学中 HOTI 的开发，声子和电路系统由于它们对这些领域的特殊贡献。从这些讨论中，我们评论了该领域的趋势和挑战以及高阶能带拓扑对其他科学学科的影响。