  We consider the minimal entropy problem, namely the question of whether there
exists a smooth metric of minimal entropy, for certain classes of 3-manifolds.
Among other resulsts, we show that if M is a closed, orientable, geometrizable
3-manifold with zero simplicial volume, then the minimal entropy can be solved
for M if and only if M admits a metric modelled on 4 of the 8 standard
3-dimensional geometries, namely $S^3$, $S^2\times R$, $E^3$, or Nil.
