  We prove the K-moduli space of cubic threefolds is identical to their GIT
moduli. More precisely, the K-(semi,poly)-stability of cubic threefolds
coincide to the corresponding GIT stabilities, which could be explicitly
calculated. In particular, this implies that all smooth cubic threefolds admit
K\"ahler-Einstein metric as well as provides a precise list of singular KE
ones. To achieve this, our main new contribution is an estimate in dimension
three of the volumes of kawamata log terminal singularities introduced by Chi
Li. This is obtained via a detailed study of the classification of three
dimensional canonical and terminal singularities, which was established during
the study of the explicit three dimensional minimal model program.
