The first comprehensive and systematic theoretical exploration of the bonding nature and energetics of the interactions between Ag(I) cation and a wide set of -ligands was accomplished. This set ranges from simple ethylene and aromatic benzene to planar and curved polyaromatic molecules and to closed-cage C-60-fullerene. Simultaneous application of two energy decomposition schemes based on different ideas, namely, NBO-NEDA and EDA-NOCV, allowed shedding light on the nature of the bonding and its energetics. Importantly, our results unambiguously indicate that reliable results can be obtained only if using more than one theoretical approach. All methods clearly revealed the importance and even domination of the ionic contribution of the bonding in all adducts, except for those of C-60-fullerene, in which the covalent component was found to be the largest. Subsequent decomposition of the orbital term onto components showed that it consists of two major parts: (i) ligand-to-metal ((C=C)s(Ag), LM) and (ii) metal-to-ligand (ML) terms, with significant domination of the former. Interestingly, while the LM component is essentially the same for all systems considered, the nature of the ML one depends on the coordination site of the polycyclic aromatic hydrocarbons (PAH). In most of adducts, the ML can be described as d(xy)(Ag)(*)(C=C) donation, whereas for systems [Ag-spoke-C12H8](+) and [Ag-spoke-C20H10](+) it corresponds to the d(z)(2)(Ag)(*)(C=C) type of interaction. As a result, the coordination mode in such complexes is switched from (2)-type to (1). Thus, the nature of the bonding, its energetics and even coordination mode in adducts of unsaturated hydrocarbons with late transition metal cations should be considered as a function of many components, which primarily includes the topology and aromaticity of the (poly)aromatic molecules.