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.