In 1939, Frenkel and Kontorova proposed a model for the motion of a dislocation (an imperfection in a crystal). The model is simple, a chain of atoms following Newton’s equation of motion. The atoms interact with their nearest neighbours via a harmonic spring and are exposed to a periodic (non-convex) on-site potential. Despite the simplicity, the model has proved to be a mathematical challenge. Iooss and Kirchg\”assner made a fundamental contribution regarding the existence of small solutions. This talk will disucss the model and the method used by Iooss and Kirchg\”assner, and then present results for the coherent motion of dislocations. The latter results are joint work with Boris Buffoni (Lausanne) and Hartmut Schwetlick (Bath).