Abstract: in this talk we focus on one-dimensional torus actions on complex smooth projective varieties of arbitrary dimension. On one hand, we introduce some combinatorics tools which we need in our situation, on the other hand we discuss how Birational Geometry takes his role. Then we introduce some classification results for special varieties called "of small bandwidth". As an application, we will discuss some consequences in the framework of contact Fano varieties of high dimension. This is a joint work with G. Occhetta, L. Sola' Conde and J. Wisniewski.