One way to measure the complexity of a smooth manifold M is to consider its minimal volume, denoted by MinVol, introduced by Gromov, which is simply defined as the infimum of the volume among metrics with sectional curvature between -1 and 1. I will introduce a variant of MinVol, called the essential minimal volume, defined as the infimum of the volume over a closure of the space of metrics with sectional curvature between -1 and 1. I will discuss the main properties of this invariant, and present estimates for negatively curved manifolds, Einstein 4-manifolds and most complex surfaces.