  This is a comprehensive review of the strong-interaction limit of density
functional theory. It covers the derivation of the limiting strictly correlated
electrons (SCE) functional from exact Hohenberg-Kohn DFT, basic aspects of SCE
physics such as the nonlocal dependence of the SCE potential on the density,
equivalent formulations and the mathematical interpretation as optimal
transport with Coulomb cost, rigorous results (including exactly soluble
cases), approximations, numerical methods, integration into Kohn-Sham DFT (KS
SCE), and applications to molecular systems, an example being that KS SCE,
unlike the local density approximation or generalized gradient approximations,
dissociates H$_2$ correctly. We have made an effort to make this review
accessible to a broad audience of physicists, chemists, and mathematicians.
