The classical Hardy–Littlewood–Sobolev theory establishes fundamental mapping properties for integral operators of potential type. It provides precise boundedness conditions for Riesz potentials acting between Lebesgue spaces, based on the order of the potential, the space dimension, and the integrability exponent. The present research develops a methodological framework for extending this theory beyond its classical limits, as demonstrated through a series of recently obtained results on Riesz potential-type operators. In particular, boundedness is studied not only in classical Lebesgue spaces but also in their modern extensions, such as grand Lebesgue spaces. These spaces form a refined scale for describing integrability, especially for functions with borderline singularities. A key focus lies in the case where the classical Sobolev condition is violated. It is shown that under certain parameters, the Riesz potential-type operator with a power–logarithmic kernel remains bounded from an $L^p$ space to a generalized Hölder space, even when the order of the potential exceeds the classical critical exponent. In this setting, the image of an integrable function is proven to possess a quantified generalized Hölder smoothness. The fundamental methodological contribution of this research is the development of a mapping theory for generalized function spaces, namely, grand Lebesgue spaces and generalized Hölder spaces. The techniques used include spectral analysis via Fourier–Laplace multipliers, Zygmund-type estimates for the continuity modulus, and the properties of grand Lebesgue spaces. This talk aims to outline how such a methodology paves the way for new, more general forms of the Hardy–Littlewood–Sobolev theory, with potential implications for problems in mathematical physics and fractional calculus.