We review the studies of a wide range of optical phenomena resulting from near-field coupling between excitons and localized surface plasmon-polaritons in hybrid nanostructures. Modern physical approaches and theoretical models reported here for the description of light absorption, scattering and extinction spectra are appropriate for interpreting physical effects in nanosystems containing metals and various excitonic materials, such as molecular aggregates of organic dyes or inorganic quantum-confined semiconductor structures. Using the example of hybrid nanosystems composed of a metallic core and an outer shell of dye J-aggregate, we perform a theoretical analysis of the optical spectra behavior in the regimes of weak, strong and ultrastrong plasmon-exciton coupling. We consider resonance and antiresonance phenomena induced by the coupling of an exciton with dipole and multipole plasmons, including a pronounced dip in light absorption, as well as the spectral band replication effect of plexcitonic nanoparticles and their dimers. In addition, we discuss the significant roles of the size-dependent permittivity of the metallic core, the effects of anisotropy and chirality of the excitonic J-aggregate shell, and the influence of an intermediate passive layer on the formation of the optical spectra of bilayer, trilayer, and multilayer nanoparticles. The review outlines the experimental and theoretical results for hybrid nanosystems of various geometrical shapes, sizes and compositions, broadens our understanding of the physical phenomena caused by the plasmon-exciton coupling, and represents the current state of research in the optics of metalorganic nanostructures.