  Topological phases of matter are ubiquitous in crystals, but less is known
about their existence in amorphous systems, that lack long-range order. In this
perspective, we review the recent progress made on theoretically defining
amorphous topological phases and the new phenomenology that they can open. We
revisit key experiments suggesting that amorphous topological phases exist in
both solid-state and synthetic amorphous systems. We finish by discussing the
open questions in the field, that promises to significantly enlarge the set of
materials and synthetic systems benefiting from the robustness of topological
matter.
