This talk is about topological and combinatorial properties of finite rank minimal systems. We establish a clear connection with the S -adic subshifts and provide necessary and sufficient conditions for a subshift to be of finite rank. Using these conditions we study the number of asympototic components of a finite rank subshift and show that there is a rank two subshift with non superlinear complexity. I will also mention results concerning the automorphism group of a finite rank subshift. This is work in progress with Fabien Durand, Alejandro Maass and Samuel Petite.