Abstract: Neutron stars are considered to be exceptional laboratories to probe the physics of matter at supranuclear densities, while the astrophysical relevance of the enigmatic objects such as black holes emerges particularly from the need of testing general relativistic predictions in strong-field regime. Indeed, the strongest gravitational fields are expected to be in the vicinity of both neutron stars and stellar-mass black holes in X-ray binaries with rather very large spacetime curvatures as compared to any other object observed in the universe. This talk is mainly devoted to the basic motivation for understanding the astrophysics of neutron stars and stellar-mass black holes to reveal the fingerprints of fundamental physics of dense matter and strong gravity through modelling of high-frequency quasi-periodic oscillations (QPOs) observed in the X-ray power spectra of these compact objects. Within a plausible scenario, we will come up with a possible explanation for the origin of QPOs with the smallest timescales. Neutron stars and black holes in low-mass X-ray binaries accrete matter from their normal companion stars. In accordance with the nature of the compact object, using different boundary conditions at the region of interaction with matter, we will argue in favor of a single model accounting for the observed correlations of high-frequency QPOs. The model simulations of the observed QPO-frequency correlations may provide us with reasonable estimates for the masses, radii, and spins of the compact objects and therefore for the state of ultradense matter and the dynamical effects of strong-field gravity.