In applying the properties of Bernoulli trials, the parameters are usually unknown and must be estimated from samples. This thesis in concerned with the design of an appropriate sampling plan or stopping rule and the construction of estimates for such purpose. The problem of estimating parameters in three models is discussed assuming dependence relations between successive trials. Various sampling plans which are well known or have potential applications are unified into a generalized sampling plan. Methods of constructing estimates are developed which can be applied to models with both dependent and independent trials. For each model, sufficient statistics, probability distributions, moments, and estimates are obtained under the generalized sampling plan. Results for various sampling plans can be derived as special cases. The properties of proposed estimates are compared with those of conventional unbiased estimates or maximum`` likelihood estimates. Finally, under given parameter values, the relative efficiencies of the various sampling plans are compared with respect to expected sample sizes and mean squared errors or variances of the estimates.