Department of MathematicsOne way to measure the complexity of a smooth manifold 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 close variant of MinVol, called the essential minimal volume, which has most of the ``good'' properties of MinVol and has also some additional advantages: it is always achieved by Riemannian metrics, which in some sense generalize hyperbolic metrics, moreover it can be estimated for Einstein 4-manifolds and most complex surfaces in terms of topology.