The intermolecular and intramolecular noncovalent interactions involving p-aromatic compounds have attracted increasing attention over the last decades in chemistry, biology and material sciences. In this review, we discuss contemporary computational studies on the nature and strength of H-p, p-p, and anion-p interactions. We emphasize how modern quantum theoretical approaches ahead of experiment can provide insight into the design of new materials and devices by tuning the p-interactions in cooperative and competitive manners. Usefulness of such approaches towards designing new materials is demonstrated with some examples of molecular recognition/sensing, self-assembly/engineering, receptors, catalysts, supramolecules, graphene, and other two-dimensional (2D) materials/devices.