K-theory for C*-algebras has proven to be a useful tool for the classification of topological phases for non-interacting fermions but also metamaterials based on classical physics. It assigns topological invariants to spectral gaps of topological insulators as well as to their surface modes and provides a robust bulk-boundary correspondence. Some of those invariants are particularly robust and admit index theorems which can be used, for example, to define topological invariants also for less regular systems such as strongly disordered insulators or Dirac/Weyl-semimetals. This talk gives an elementary introduction and presents some recent results.
