We investigate the following question: When a group G admits a pseudocompact Hausdorff group topology? E. K. van Douwen was the first who found same constraints for the existence of such a topology. We find some other constraints and also a lot of sufficient conditions under which a group does admit a pseudocompact Hausdorff group topology. These sufficent conditions almost coincide with necessary ones for four classes of groups, free groups, free Abelian groups, torsion Abelian groups and divlsable Abelian groups. The gap here depends often on some extra set-theoretic assumption which is known to be equiconsistent with the existence of large cardinals.