In many empirical applications of regression discontinuity designs, the running variable used by the administrator to assign treatment is only observed with error. This paper provides easily interpretable conditions under which ignoring the measurement error nonetheless yields an estimate with a causal interpretation: the average treatment effect for units with the value of the observed running variable equal to the cutoff. To accommodate various types of measurement error, we propose to conduct inference using recently developed bias-aware methods, which remain valid even when discreteness or irregular support in the observed running variable may lead to partial identification. We illustrate the results for both sharp and fuzzy designs in an empirical application.