Abstract The critical dimensions of the sigma model formulations of string theories are calculated by demanding unitarity in one loop amplitudes. For both the open and closed bosonic strings it is verified that d = 26 − d G /(1 + 1 2 C A in agreement with the k = 1 result obtained from Lorentz invariance of the free string theory. Furthermore, it is shown that modular invariance of the closed bosonic string restricts the sigma model group G to be selfdual. Similar results are obtained for the fermionic string models; in particular it is found that d = 10 − 2 3 − d G /(1 + 1 2 C A ) − 1 3 d G . The together with selfduality of G are inconsistent for the closed fermionic string and lead to the following cases for the closed bosonic string: (i) d = 18, G = E 8; (ii) d = 10, G = E 8 × E 8or Spin (32)/Z 2; (iii) d = 2, G = one of the groups corresponding to the 24 even selfdual lattices in 24 dimensions.