### Abstract

Aperiodically ordered patterns, i.e., non-repeating yet highly structured tilings such as the Penrose tiling(s), have proved to be a source of interaction between noncommutative geometry, dynamics and topology. Associated K-theoretic invariants have given both geometric and physical information about the underlying patterns and their realisation as models for so-called 'quasicrystals'. This talk will present both background and some recent results and perspectives aiming to understand further the properties of aperiodic patterns, their K-theory and related topology.