In this paper, we develop a method to assess the sensitivity of local average treatment effect estimates to potential violations of the monotonicity assumption of Imbens and Angrist (1994). We parameterize the degree to which monotonicity is violated using two sensitivity parameters: the first one determines the share of defiers in the population, and the second one measures differences in the distributions of outcomes between compliers and defiers. For each pair of values of these sensitivity parameters, we derive sharp bounds on the outcome distributions of compliers in the first-order stochastic dominance sense. We identify the robust region that is the set of all values of sensitivity parameters for which a given empirical conclusion, e.g. that the local average treatment effect is positive, is valid. Researchers can assess the credibility of their conclusion by evaluating whether all the plausible sensitivity parameters lie in the robust region. We obtain confidence sets for the robust region through a bootstrap procedure and illustrate the sensitivity analysis in an empirical application. We also extend this framework to analyze treatment effects of the entire population.