Author(s): Tirth Shah, Christian Brendel, Vittorio Peano, and Florian Marquardt Artificially engineered mechanical systems, sometimes called metamaterials, offer many promising applications on length scales ranging from macroscopic systems to the nanoscale. A topic of particular interest is the existence of topologically protected phononic edge states in such systems that are analogous to the electronic edge states that give rise to the quantum Hall effect. This Colloquium gives an introduction to topologically protected transport in metamaterials and its applications for controlling acoustic transport. [Rev. Mod. Phys. 96, 021002] Published Thu Apr 18, 2024