Curie temperatures (TC) of several diluted magnetic semiconductors are calculated from first-principles by using a mapping on a Heisenberg model in a mean field approximation. Very large TC’s are obtained for V- or Cr-doped ZnSe and ZnTe and Cr-doped GaN, GaP and GaAs. Effects of additional carrier doping into (Ga, Mn)As and (Ga, Mn)N are investigated. TC of (Ga, Mn)N increases by hole doping, while (Ga, Mn)As is insensitive to hole doping. The origin of the ferromagnetism in Mn-doped III–V DMS is also discussed. It is found that if impurity bands are formed in the gap, as it is the case for (Ga, Mn)N, double exchange dominates leading to a characteristic √c dependence of TC as a function of the Mn concentration c. On the other hand, if the d-states are localized, as in (Ga, Mn)Sb, Zener’s p–d exchange prevails resulting in a linear c-dependence of TC. (Ga, Mn)As is an intermediate case, showing a √c like behavior in the Local density approximation (LDA), but a nearly linear c-dependence, if the LDA+U method, with U=4eV, is used.