  Charge transfer plays a crucial role in many processes of interest in
physics, chemistry, and bio-chemistry. In many applications the size of the
systems involved calls for time-dependent density functional theory (TDDFT) to
be used in their computational modeling, due to its unprecedented balance
between accuracy and efficiency. However, although exact in principle, in
practise approximations must be made for the exchange-correlation functional in
this theory, and the standard functional approximations perform poorly for
excitations which have a long-range charge-transfer component. Intense progress
has been made in developing more sophisticated functionals for this problem,
which we review. We point out an essential difference between the properties of
the exchange-correlation kernel needed for an accurate description of
charge-transfer between open-shell fragments and between closed-shell
fragments. We then turn to charge-transfer dynamics, which, in contrast to the
excitation problem, is a highly non-equilibrium, non-perturbative, process
involving a transfer of one full electron in space. This turns out to be a much
more challenging problem for TDDFT functionals. We describe dynamical step and
peak features in the exact functional evolving over time, that are missing in
the functionals currently used. The latter underestimate the amount of charge
transferred and manifest a spurious shift in the charge transfer resonance
position. We discuss some explicit examples.
