The thermophysical properties of fluids near their liquid-vapor critical point are governed by universal critical phenomena, formalized theoretically after the works of Kaddanoff, Widom and Wilson in the early seventies and afterwards. But the influence of these peculiar properties on the dynamics of fluids close to the critical point is a relatively new subject, which finds its early developments around 1990. In this talk, we will present how the paradoxical nature of such fluids, intermediate between gases and liquids, leads to complex and before-unexpected thermal and mechanical responses, among which a fourth mode of heat transfer still unknown some 15 years ago. We will emphasize the benefit brought by asymptotic and stability analysis to that kind of studies, through the examples of both microgravity and gravity-driven heat and mass transfer. Beyond the wide range of industrial applications based on supercritical fluids to which these results may apply, we will stress how small-scale near-critical fluid cells tend to behave like large geophysical fluid systems. This analogy opens up the future possibility of using near-critical fluids for small-scale laboratory experiments mimicking the behavior of large oceanic or atmospheric phenomena.