Simplicial volume is a homotopy invariant of closed manifolds - defined in terms of the singular chain complex - which measures the efficiency of representing the fundamental class by singular chains. So it gives an indication of how difficult it is to triangulate the manifold in question. It was first introduced by Gromov in the early 1980's and since then it has been central to the investigation of the relationship between the large scale geometry and the topology of manifolds. The most fundamental theorem in this subject is Gromov's Main Inequality which bounds the simplicial volume of a manifold M by its Riemannian volume provided M satisfies a lower Ricci curvature bound. After a short introduction to simplicial volume, I will address new developments around curvature-free versions of the main inequality.