  This is a short (and somewhat informal) contribution to the proceedings of
the XVIth International Congress on Mathematical Physics, Prague, 2009, written
up by the second author. We describe how the recent proof of the existence and
conformal covariance of the scaling limits of dynamical and near-critical
planar percolation implies the existence and several topological properties of
the scaling limit of the Minimal Spanning Tree, and that it is invariant under
scalings, rotations and translations. However, we do not expect conformal
invariance: we explain why not and what is missing for a proof.
