Topological phases are remarkable because of their propagating surface
states resistant to disorder and their universally quantised responses to
electromagnetic fields. Topological phenomena are by now ubiquitously
observed in crystals, both in solids and synthetic systems. In contrast,
topology in non-crystalline matter, such as amorphous or
quasicrystalline materials, is much harder to predict, diagnose and
discover. This is because crystalline periodicity is central to how we
understand topological materials, although it is not necessary to define
topological phases. In this talk I will present recent theoretical progress
and experimental signatures of amorphous topological matter. I will
discuss tools to predict them, and their potential to host novel and
controllable topological phenomena both in amorphous metals and
insulators.