Fluid interfaces with nanoscale radii of curvature are generating great interest, both for their applications and as tools to probe our fundamental understanding. One important question is what is the smallest radius of curvature at which the three main thermodynamic combined equilibrium equations are valid: the Kelvin equation for the effect of curvature on vapor pressure, the Gibbs-Thomson equation for the curvature-induced freezing point depression, and the Ostwald-Freundlich equation for the curvature-induced increase in solubility. The objective of this Perspective is to provide conceptual, molecular modeling, and experimental support for the validity of these thermodynamic combined equilibrium equations down to the smallest interfacial radii of curvature. Important concepts underpinning thermodynamics, including ensemble averaging and Gibbs's treatment of bulk phase heterogeneities in the region of an interface, give reason to believe that these equations might be valid to smaller scales than was previously thought. There is significant molecular modeling and experimental support for all three of the Kelvin equation, the Gibbs-Thomson equation, and the Ostwald-Freundlich equation for interfacial radii of curvature from 1 to 4 nm. There is even evidence of sub-nanometer quantitative accuracy for the Kelvin equation and the Gibbs-Thomson equation.