We investigate exciton ground states in Si and 3C-SiC quantum dots by using the effective mass theory, taking account of the conduction- and valence-band mass anisotropy as well as the small spin-orbit splitting energy. The degenerate hole and exciton states are partly split by the mass anisotropy. The anisotropy splitting energies in quantum dots are different dramatically from their bulk value due to quantum size effects. The assumed changeable spin-orbit splitting energy may change the ordering of the anisotropy-split energy levels. Taking account of the exchange interaction, the degeneracy of the exciton states is further lifted. Due to the anisotropy and exchange splitting, the 48-fold exciton ground state will be split into two 18-fold triplets and two 6-fold singlets. The lowest three states are optically forbidden for Si quantum dots, which leads to a Stokes shift of luminescence. The theroretical shift agrees well with the experimental data. Furthermore, the exciton band gap and binding energy as a function of dot radius are presented both for Si and for 3C-SiC quantum dots. The band gap of Si quantum dots agrees well with the recent photoluminescence results of size-separated quantum dots by Ledoux et al. and absorption data of Furukawa et al.