  We present a review of recent theoretical results concerning the many-body
localization (MBL) phenomenon, with the emphasis on dynamical density
correlations and transport quantities. They are shown to be closely related,
providing a comprehensive description of the ergodic-to-nonergodic transition,
consistent with experimental findings. While the focus is set mostly on the
one-dimensional model of interacting spinless fermions, we also present
evidence for the absence of full MBL in the one-dimensional Hubbard model and
for the density-wave decay induced by the inter-chain coupling.
