  We survey the definitions and some important properties of several asymptotic
invariants of smooth manifolds, and discuss some open questions related to
them. We prove that the (non-)vanishing of the minimal volume is a
differentiable property, which is not invariant under homeomorphisms. We also
formulate an obstruction to the existence of Einstein metrics on four-manifolds
involving the volume entropy. This generalizes both the Gromov--Hitchin--Thorpe
inequality and Sambusetti's obstruction.
