This paper shows nonparametric identification of quantile treatment effects (QTE) in the regression discontinuity design. The distributional impacts of social programs such as welfare, education, training programs and unemployment insurance are of large interest to economists. QTE are an intuitive tool to characterize the effects of these interventions on the outcome distribution. We propose uniformly consistent estimators for both potential outcome distributions (treated and non-treated) for the population of interest as well as other function-valued effects of the policy including in particular the QTE process. The estimators are straightforward to implement and attain the optimal rate of convergence for one-dimensional nonparametric regression. We apply the proposed estimators to estimate the effects of summer school on the distribution of school grades, complementing the results of Jacob and Lefgren (2004).