In this paper, we investigate what can be learned about average counterfactual outcomes when it is assumed that treatment response functions are smooth. The smoothness conditions in this paper amount to assuming that the di fferences in average counterfactual outcomes are bounded under different treatments. We obtain a set of new partial identification results for the average treatment response by imposing smoothness conditions alone, by combining them with monotonicity assumptions, and by adding instrumental variables assumptions to treatment responses. We give a numerical illustration of our findings by reanalyzing the return to schooling example of Manski and Pepper (2000) and demonstrate how one can conduct sensitivity analysis by varying the degrees of smoothness assumption. In addition, we discuss how to carry out inference based on the existing literature using our identication results and illustrate its usefulness by applying one of our identification results to the Job Corps Study dataset. Our empirical results show that there is strong evidence of the gender and race gaps among the less educated population.