Atmospheric rivers (ARs) generate most of the economic losses associated with flooding in the western United States and are projected to increase in intensity with climate change. This is of concern as flood damages have been shown to increase exponentially with AR intensity. To assess how AR-related flood damages are likely to respond to climate change, we constructed county-level damage models for the western 11 conterminous states using 40 years of flood insurance data linked to characteristics of ARs at landfall. Damage functions were applied to 14 CMIP5 global climate models under the RCP4.5 "intermediate emissions" and RCP8.5 "high emissions" scenarios, under the assumption that spatial patterns of exposure, vulnerability, and flood protection remain constant at present day levels. The models predict that annual expected AR-related flood damages in the western United States could increase from $1 billion in the historical period to $2.3 billion in the 2090s under the RCP4.5 scenario or to $3.2 billion under the RCP8.5 scenario. County-level projections were developed to identify counties at greatest risk, allowing policymakers to target efforts to increase resilience to climate change.