The study of nanostructured artificial media for optics has expanded rapidly over the last few decades, coupled to improvements of fabrication technology that has enabled investigation of previously unrealisable optical scattering systems. Such development is complemented by renewed impetus to understand the physics of optical scattering from complex subwavelength geometry and nanoparticle systems. Here I investigate specifically the optical properties of closely-packed arrangements of nanoparticles, known as nanoparticle oligomers, which provide an intuitive platform for analytical and numerical study on the formation and interplay of collective resonances. I consider both plasmonic nanoparticles, but also high-refractive-index dielectric nanoparticles that support Mie-type electric and magnetic dipole resonances. Specific outcomes of this study are listed as follows. (i)~A new model is presented for optical Fano resonances, which is based on interference between nonorthogonal eigenmodes of the associated scattering object. This is demonstrated to correctly model Fano resonances in both plasmonic and high-refractive-index dielectric nanoparticle oligomers; it also revealed capacity for two-channel Fano interference in the magnetic dipolar response from the dielectric oligomers. (ii)~Polarisation-independent scattering and absorption losses are shown to be enforced by $n$-fold discrete rotational symmetry, $C_n$ ($n\geq3$), and reciprocal degeneracy of eigenmodes. (iii)~A new form of circular dichroism is presented, which occurs due to the interaction of nonorthogonal resonances, and impacts the ratio of radiative scattering loss to dissipative absorption loss in reciprocal plane waves. Geometric asymmetry and optical chirality is also reviewed to quantify the minimum symmetries that must be broken to allow other circular dichroism effects in chiral and achiral scattering objects. The sequence of general theoretical conclusions (i)-(iii) serve to build the understanding of optical scattering from nanoparticle systems while removing existing ambiguities.