Nonregular designs are used widely in experiments due to their run size economy and flexibility. These designs include the Plackett-Burman designs and many other symmetrical and asymmetrical orthogonal arrays. Supersaturated designs have become increasingly popular in recent years because of the potential in saving run size and its technical novelty. In this paper, a novel combinatorial criterion, called minimum moment aberration, is proposed for assessing the goodness of nonregular designs and supersaturated designs. The new criterion, which is to sequentially minimize the power moments of the number of coincidence among runs, is a good surrogate with tremendous computational advantages for many statistically justified criteria, such as minimum G2-aberrration, generalized minimum aberration and E(s2). In addition, the minimum moment aberration is conceptually simple and convenient for theoretical development. The general theory developed here not only unifies several separate results, but also provides many novel results on nonregular designs and supersaturated designs.