We use the symmetry arguments of Laudau and Lifshitz, described in the preceding papers, to derive the Landau-Ginzburg-Wilson Hamiltonians corresponding to several antiferromagnetic systems. The phase transitions associated with these models are then studied using the exact renormalization-group technique in $d=4\ensuremath{-}\ensuremath{\epsilon}$ dimensions. We find that the fcc type-III antiferromagnets $\stackrel{\ensuremath{\rightarrow}}{\mathrm{m}}\ensuremath{\perp}[100],\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}=(\frac{1}{2},0,1)$ and the spiral magnets Eu and Cr are described by two $n=12$ models. For both models, the renormalization-group recursion relations have no stable fixed points. This might explain the existence of first-order transitions in Cr and Eu. We also find that the antiferromagnet Mn${\mathrm{S}}_{2}$ is described by the $n=6$ model derived by Mukamel and Krinsky to represent Tb${\mathrm{D}}_{2}$, Nd, and ${\mathrm{K}}_{2}$Ir${\mathrm{Cl}}_{6}$. Since this model has one stable fixed point, it is predicted that the four compounds belong to the same universality class. Similarly, the spiral magnets Tb, Dy, and Ho correspond to the $n=4$ model which was used to describe Nb${\mathrm{O}}_{2}$, Tb${\mathrm{Au}}_{2}$, and Dy${\mathrm{C}}_{2}$, and it is predicted that they all have the same critical behavior. Existing experimental data are discussed and several experiments are suggested.