In this talk I will explain the notion of topological non-trivial band structures in real materials. In the first part I will give examples of parallel transport and holonomies in classical and quantum physics. By relating these concepts to the electronic structure of periodic solids, I will then explain topological non-trivial states of matter such as topological insulators as well as Weyl and Dirac metals. Furthermore computational challenges as well as open problems regarding these topics will be discussed.