Metric inequalities on Riemannian manifolds subject to lower scalar curvature bounds have become an important aspect in the study of scalar curvature in the recent years, in partic-ular due to conjectures raised by Gromov. This has lead to new geometric phenomena involving the scalar curvature, but which are vaguely reminiscent of classical results in comparison ge-ometry for sectional or Ricci curvature. In this talk, I will give an introduction to this type of questions and survey some of the recent results.