Concentration-response (C-R) functions relating concentrations of pollutants in ambient air to mortality risks or other adverse health effects provide the basis for many public health risk assessments, benefits estimates for clean air regulations, and recommendations for revisions to existing air quality standards. The assumption that C-R functions relating levels of exposure and levels of response estimated from historical data usefully predict how future changes in concentrations would change risks has seldom been carefully tested. This paper critically reviews literature on C-R functions for fine particulate matter (PM2.5) and mortality risks. We find that most of them describe historical associations rather than valid causal models for predicting effects of interventions that change concentrations. The few papers that explicitly attempt to model causality rely on unverified modeling assumptions, casting doubt on their predictions about effects of interventions. A large literature on modern causal inference algorithms for observational data has been little used in C-R modeling. Applying these methods to publicly available data from Boston and the South Coast Air Quality Management District around Los Angeles shows that C-R functions estimated for one do not hold for the other. Changes in month-specific PM2.5 concentrations from one year to the next do not help to predict corresponding changes in average elderly mortality rates in either location. Thus, the assumption that estimated C-R relations predict effects of pollution-reducing interventions may not be true. Better causal modeling methods are needed to better predict how reducing air pollution would affect public health.