What is now called "resonance theory" has a long and conflicted history. We first sketch the early roots of resonance theory, its heritage of diverse physics and chemistry conceptions, and its subsequent rise to reigning chemical bonding paradigm of the mid-20th century. We then outline the alternative "natural" pathway to localized Lewis- and resonance-structural conceptions that was initiated in the 1950s, given semi-empirical formulation in the 1970s, recast in ab initio form in the 1980s, and successfully generalized to multi-structural "natural resonance theory" (NRT) form in the 1990s. Although earlier numerical applications were often frustrated by the ineptness of then-available numerical solvers, the NRT variational problem was recently shown to be amenable to highly efficient convex programming methods that yield provably optimal resonance weightings at a small fraction of previous computational costs. Such convexity-based algorithms now allow a full "reboot" of NRT methodology for tackling a broad range of chemical applications, including the many familiar resonance phenomena of organic and biochemistry as well as the still broader range of resonance attraction effects in the inorganic domain. We illustrate these advances for prototype chemical applications, including (i) stable near-equilibrium species, where resonance mixing typically provides only small corrections to a dominant Lewis-structural picture, (ii) reactive transition-state species, where strong resonance mixing of reactant and product bonding patterns is inherent, (iii) coordinative and related supramolecular interactions of the inorganic domain, where sub-integer resonance bond orders are the essential origin of intermolecular attraction, and (iv) exotic long-bonding and metallic delocalization phenomena, where no single "parent" Lewis-structural pattern gains pre-eminent weighting in the overall resonance hybrid.