  Time-dependent density functional theory has emerged as a method of choice
for calculations of spectra and response properties in physics, chemistry, and
biology, with its system-size scaling enabling computations on systems much
larger than possible otherwise. While increasingly complex and interesting
systems have been successfully tackled with relatively simple functional
approximations, there has also been increasing awareness that these functionals
tend to fail for certain classes of approximations. I review the fundamental
challenges the approximate functionals have in describing double-excitations
and charge-transfer excitations, which are two of the most common impediments
for the theory to be applied in a black box way. At the same time, I describe
the progress made in recent decades in developing functional approximations
that give useful predictions for these excitations.
