Originally discovered in condensed matter systems, topological insulators (TIs) have been ubiquitously extended to various fields of classical wave physics including photonics, phononics, acoustics, mechanics, and microwaves. In the bulk, like any other insulator, electronic TIs exhibit an excessively high resistance to the flow of mobile charges, prohibiting metallic conduction. On their surface, however, they support one-way conductive states with inherent protection against certain types of disorder and defects, defying the common physical wisdom of electronic transport in presence of impurities. When transposed to classical waves, TIs open a wealth of exciting engineering-oriented applications, such as robust routing, lasing, signal processing, switching, etc., with unprecedented robustness against various classes of defects. In this article, we first review the basic concept of topological order applied to classical waves, starting from the simple one-dimensional example of the Su–Schrieffer–Heeger (SSH) model. We then move on to two-dimensional wave TIs, discussing classical wave analogues of Chern, quantum Hall, spin-Hall, Valley-Hall, and Floquet TIs. Finally, we review the most recent developments in the field, including Weyl and nodal semimetals, higher-order topological insulators, and self-induced non-linear topological states.