Summary General estimable parameters in randomized response models have been characterized as functionals of several independent distributions for which the so-called generalized U-statistics are known to be optimal unbiased estimators. The object of the present paper is to determine the individual sample sizes in such a way that the efficiency of these generalized U-statistics is maximized in some sense. Specifically, along the general lines of Williams and Sen (1973, 1974), the problem of sequential estimation of general parameters in randomized response models is formulated and some applications to live data are made. selects one of the two questions at random without revealing the outcome to the interviewer. The probabilities of selecting the two questions are set beforehand. Thus, the randomization affords protection to the respondent and hence potential embarrassments and stigma for answering a sensitive question are avoided, and therefore, the primary reason for either a refusal or an evasive answer does not exist. In a recent paper (Sen, 1974), the present author has formulated the theory of estimation of general estimable parameters in a RRM. It turns out that such estimable parameters (of individual or composite distributions) are functionals of the two randomized response distributions. This characterization enables one to make use of the theory of generalized U-statistics and von Mises' differentiable statistical functions for studying the general properties of natural estimators in RRM. The current investigation attempts to provide sequential solutions to some related problems of estimation in RRM. Two basic problems are considered here. First, for the problem of point estimation of one or more estimable parameters in a RRM, one is naturally interested in choosing the individual sample sizes in such a manner that the precision of the estimators is maximized in some sense. Since the covariance matrix of the individual kernels is not generally known, no fixed sample size procedure sounds feasible. A sequential procedure based on sequentially updated versions of estimators of this covariance matrix is developed here, and this procedure is optimal in a certain sense. Second, for unknown covariance matrices, a fixed-width confidence interval for an estimable parameter in a RRM can also be achieved only in a sequential setup. In this context, the problem of determining the individual sample sizes so as to maximize the efficiency (or minimize the average sample number of the total sample) is considered.