In this talk, I will discuss the future-global nonlinear behavior of relativistic fluids evolving in expanding spacetimes. I will focus on the ways in which the global behavior of the fluid is affected by both the spacetime expansion rate and the fluid equation of state. These topics are physically relevant for the following reasons: i) In cosmology, the relativistic fluid model is the most often used model for the “normal” matter content of our physical spacetime. ii) Experimental evidence indicates that our spacetime is undergoing accelerated expansion. iii) The precise expansion rate is not known, and is in fact a current topic of debate. iv) Prior mathematical results show that the fluid’s global behavior is heavily influenced by the expansion rate. For example, in Minkowski spacetime (which is expansion-free), D. Christodoulou showed that the constant fluid solutions with positive energy density are unstable. More precisely, he exhibited an open set of data such that a) it contains arbitrarily small perturbations of the constant states, and b) the corresponding solutions necessarily develop shock singularities in finite time. In contrast, when the spacetime is exponentially expanding and the fluid is irrotational, I. Rodnianski and I showed that under some often-made assumptions on the equation of state, the spatially constant fluid solutions with positive energy density are future-stable. This talk concerns the expansion rates in between 0 and exponential without the irrotationality assumption. Furthermore, I will discuss several new results for the “pressureless dust” equation of state p = 0 and the “radiation” equation of state p = (1/3)ρ, which play fundamental roles in cosmology.