Character table and blocks of finite simple triality groups $\sp 3D\sb 4(q)$

Based on recent work of Spaltenstein [14] and the Deligne-Lusztig theory of irreducible characters of finite groups of Lie type, in this paper the character table of the finite simple groups ${}^3{D_4}(q)$ is given. As an application we obtain a classification of the irreducible characters of ${}^3{D_4}(q)$ into $r$-blocks for all primes $r > 0$. This enables us to verify Brauer's height zero conjecture, his conjecture on the bound of irreducible characters belonging to a give block, and the Alperin-McKay conjecture for the simple triality groups ${}^3{D_4}(q)$. It also follows that for every prime $r$ there are blocks of defect zero in ${}^3{D_4}(q)$.