  Stimulated by the recent experimental progress on the $T_{\psi\psi}$ states,
the fully charmed tetraquark spectroscopy is systemically investigated by dint
of the Godfrey-Isgur relativized diquark model, the modified Godfrey-Isgur
relativized diquark model with the color screening effects, and the
nonrelativistic diquark model. The theoretical results of the
diquark-antidiquark scenarios propose to interpret the $T_{\psi\psi}(6200)$,
$T_{\psi\psi}(6600)$, $T_{\psi\psi}(6900)$, and $T_{\psi\psi}(7300)$ structures
as the candidates of the $1S$-wave, $1P/2S$-wave, $1D/2P$-wave, and
$2D/3P/4S$-wave fully charmed tetraquark states, respectively. On account of
the deficiency of sufficient experimental information, e.g., the parities of
the newly observed $T_{\psi\psi}$ states, there are uncertainties about the
assignments of the $T_{\psi\psi}(6600)$, $T_{\psi\psi}(6900)$, and
$T_{\psi\psi}(7300)$ states. It is demonstrated that the further experimental
survey on the $cc\bar c\bar c$ states, implemented by the LHCb, ATLAS, CMS, and
other collaborations, ought to be continued in the future.
