In recent years, there has been great interest in the use of adaptive features in clinical trials (i.e., changes in design or analyses guided by examination of the accumulated data at an interim point in the trial) that may make the studies more efficient (e.g., shorter duration, fewer patients). Many statistical methods have been developed to maintain the validity of study results when adaptive designs are used (e.g., control of the Type I error rate). Group sequential designs, which allow early stopping for efficacy in light of compelling evidence of benefit or early stopping for futility when the likelihood of success is low at interim analyses, have been widely used for many years. In this dissertation, we study several aspects of statistical issues in group sequential/adaptive designs. Sample size re-estimation has drawn a great deal of interest due to its permitting revision of the target treatment difference based on the unblinded interim analysis results from an ongoing trial. A possible risk of ublinded sample size re-estimation is that the exact treatment effect being observed at interim analysis might be back-calculated from the modified sample size, which might jeopardize the integrity of the trial. In the first project, we propose a pre-specified stepwise two-stage sample size adaptation to lessen the information on treatment effect that would be revealed. We minimize expected sample size among a class of these designs and compare efficiency with the fully optimized two-stage design, optimal two-stage group sequential design and designs based on promising conditional power. In the second project, we define the complete ordering of a group sequential sample space and show that a Wang-Tsiatis boundary family or an exponential spending function family can completely order the sample space. We also propose a simple method to transform a spending function to a completely ordered sample space when using the sequential p-value ordering. This method is also extended to β-spending functions for p-values to reject the alternative hypothesis. In the third project, we propose a simple approach for controlling the familywise error rate in a group sequential design with multiple testing. We apply sequential p-values at the interim analysis from a group sequential design to the sequentially rejective graphical procedure which is based on the closure principle. We also use simulations to study the operating characteristics of multiple testing in group sequential designs. We show that in terms of expected sample size, using a group sequential design in multiple hypothesis testing is more efficient than fixed sample size designs in many scenarios.