  For a given electronic excited state, the 0-0 energy ($T_0$ or $T_{00}$) is
the simplest property allowing straightforward and physically-sound comparisons
between theory and (accurate) experiment. However, the computation of 0-0
energies with \emph{ab initio} approaches requires determining both the
structure and the vibrational frequencies of the excited state, which limits
the quality of the theoretical models that can be considered in practice. This
explains why only a rather limited, yet constantly increasing, number of works
have been devoted to the determination of this property. In this contribution,
we review these efforts with a focus on benchmark studies carried out for both
gas phase and solvated compounds. Over the years, not only as the size of the
molecules increased, but the refinement of the theoretical tools has followed
the same trend. Though the results obtained in these benchmarks significantly
depend on both the details of the protocol and the nature of the excited
states, one can now roughly estimate, in the case of valence transitions, the
overall accuracy of theoretical schemes as follows: $1$ eV for CIS,
$0.2$--$0.3$ eV for CIS(D), $0.2$--$0.4$ eV for TD-DFT when one employs hybrid
functionals, $0.1$--$0.2$ eV for ADC(2) and CC2, and $0.04$ eV for CC3, the
latter approach being the only one delivering chemical accuracy on a
near-systematic basis.
