Predicting and constraining RNA virus evolution require understanding the molecular factors that define the mutational landscapes accessible to these pathogens. RNA viruses typically have high mutation rates, resulting in frequent production of protein variants with compromised biophysical properties. Viral evolution is necessarily constrained by the consequent challenge to protein folding and function. It follows that host proteostasis mechanisms hijacked by viruses could be significant determinants of the fitness of viral protein variants, serving as a critical force shaping viral evolution. I will discuss a series of experiments designed to test this possibility by integrating chemical tools to regulate host proteostasis with both quantitative and qualitative strategies to assess the consequences for influenza and HIV evolution. Results from these studies are providing new insights into features of host–pathogen interactions that shape viral evolution, and into the potential design of host proteostasis-targeted antiviral therapeutics that are refractory to resistance.