A class of sequential designs for estimating the percentiles of a quantal response curve is proposed. Its updating rule is based on an efficient summary of all of the data available via a parametric model. The logit-MLE version of the proposed designs can be viewed as a natural analog of the Robbins—Monro procedure in the case of binary data. It is shown to be asymptotically consistent, optimal, and nonparametric via its connection with the latter procedure. For certain choices of initial designs, the proposed method performs very well in a simulation study for sample sizes up to 35. A nonparametric sequential design, via the Spearman—Kärber estimator, for estimating the median is also proposed.