The simplicial volume (or Gromov norm) is a homotopy invariant of compact manifolds introduced in 1982 by Gromov in his pioneering paper "Volume and Bounded Cohomology". Roughly speaking, the simplicial volume measures how it is difficult to describe the manifold in question in terms of real singular chains. Despite its topological meaning, the simplicial volume turns out to be a fundamental tool for understanding rigidity phenomena, i.e. to establish obstructions to the existence of some geometric structures in terms of topological invariants. More precisely, working with negatively curved manifolds, simplicial volume provides useful information about their Riemannian volume. The aim of this talk is to give an accessible overview about the notion of simplicial volume and to discuss the main techniques involved in this context (some key words are: degree of maps, amenability and negative curvature). If there will be enough time, I will discuss some classical applications like Mostow rigidity and the study of the variation of the volume during a hyperbolic Dehn filling.