Recent advances in the knowledge of heat transport near the liquid-gas critical point under the influence of the piston effect are reviewed with an emphasis on the different physical mechanisms and timescales in regard to thermal and density relaxations. Near the critical point, thermophysical properties exhibit singular behaviors, such as the diverging compressibility and vanishing thermal diffusivity. The resulting fast thermalization leads to the unexpected discovery of the piston effect. We describe the previous theoretical, numerical, and experimental investigations of this unique critical phenomenon and related topics, including its thermoacoustic nature with various nonlinear features on the acoustic timescale. Hydrodynamic and thermovibrational instabilities on the diffusion timescale in near-critical fluids are addressed as well. The review ends with a brief discussion of the merits and limitations of selected research methods in common use. •The thermoacoustic nature with nonlinear features of the piston effect is emphasized.•The interplay between the piston effect and mechanical instabilities is analyzed.•The merits and limitations of various modeling and experimental techniques are discussed.