Appears in collection : Summer School 2021 - Curvature Constraints and Spaces of Metrics

I will discuss the notion of minimal volume and some of its variants. The minimal volume of a manifold is defined as the infimum of the volume over all metrics with sectional curvature between -1 and 1. Such an invariant is closely related to "collapsing theory", a far reaching set of results developed by Cheeger, Gromov, Fukaya and others to describe bounded sectional curvature metrics. Most of my talks will be focused on presenting the main aspects of this theory: thick-thin decomposition, F-structures and N-structures, collapsing constructions... Relations of the minimal volume to topological invariants will be explained, and some open questions will be mentioned.