Phase I clinical trials are conducted to determine the dose-response curve of a new drug with respect to toxic side effects and, in particular, to estimate the maximum tolerated dose (MTD). In this paper we take a Bayesian approach to the problem of making inferences about the MTD. Working with broad classes of priors, we obtain the posterior distribution of the MTD and study its properties. We also address the question of providing updated assessments of the risk of toxicity for new patients entering the study at a specific dose level. These assessments would be useful in deciding issues of study management and ethics. Our analysis pays particular attention to the sensitivity of the inferences and risk assessments to the choice of prior and the choice of model for the dose-response relationship.