Barchan dunes are crescent-shaped, isolated sand dunes that are found in large migrating colonies in many arid zones on Earth and also on Mars. To elucidate the basic physical mechanisms underlying dune formation and migration, I will discuss a minimal model. It predicts non-trivial similarity relations for the shape and dynamics of aeolian and submarine dunes under diverse wind and sand conditions. They reveal a strong universality behind the manifold structures formed by wind-blown sand --- based on a subtle interplay of the scale invariance and spontaneous symmetry breaking of the turbulent wind, and the breaking of scale invariance by the grains.