We consider the design of single- and multiple-stage dose-response trials in which the probability of response is a logistic function of the dose. Knowledge of the parameters of the logistic at the time of planning the trial is represented by a Gaussian prior distribution. Methods are presented for determining a design that approximately optimizes a measure of the accuracy of estimation averaged over the prior distribution. Changes in design due to uncertainty of parameter values are described as well as the changes in sample size required to produce a specified precision. In multiple-stage trials, the initial stage is planned as if it were the only stage. For succeeding stages, the initial Gaussian prior distribution is updated using outcomes at the previous stages. At each such stage, the design optimizes a chosen criterion averaged over the updated prior distribution. The effectiveness of this methodology is evaluated by comparing the operating characteristics of two-stage designs with those of single-stage designs.