The chapter discusses the sequential design of experiments, a significant advancement in statistical theory that allows for the flexibility of sample size and composition based on observed data. This approach contrasts with traditional methods where the sample size and composition are fixed before experimentation begins. The introduction highlights the historical and mathematical reasons for this shift, emphasizing the benefits of sequential methods in reducing average sample sizes and improving efficiency. The text provides examples of sequential designs, such as the double sampling inspection method in industrial quality control and sequential analysis developed during World War II. It also discusses the challenges and limitations of sequential design theory, noting that it is still incomplete and requires further development. A key example is the problem of estimating the difference between two populations, where the goal is to maximize the expected value of the sum of observations. The chapter explores various sampling rules and their operating characteristics, showing that some rules can significantly improve efficiency compared to fixed rules. The chapter also addresses more complex problems, such as determining the optimal sequence of treatments in a continuum of populations and the issue of optional stopping, where the experimenter can stop sampling early to influence the results. It emphasizes the importance of robust statistical methods to guard against the effects of optional stopping and the need for empirical studies to approximate the operating characteristics of sequential designs. Overall, the chapter underscores the potential of sequential design in enhancing statistical efficiency and the ongoing efforts to develop more sophisticated methods in this area.The chapter discusses the sequential design of experiments, a significant advancement in statistical theory that allows for the flexibility of sample size and composition based on observed data. This approach contrasts with traditional methods where the sample size and composition are fixed before experimentation begins. The introduction highlights the historical and mathematical reasons for this shift, emphasizing the benefits of sequential methods in reducing average sample sizes and improving efficiency. The text provides examples of sequential designs, such as the double sampling inspection method in industrial quality control and sequential analysis developed during World War II. It also discusses the challenges and limitations of sequential design theory, noting that it is still incomplete and requires further development. A key example is the problem of estimating the difference between two populations, where the goal is to maximize the expected value of the sum of observations. The chapter explores various sampling rules and their operating characteristics, showing that some rules can significantly improve efficiency compared to fixed rules. The chapter also addresses more complex problems, such as determining the optimal sequence of treatments in a continuum of populations and the issue of optional stopping, where the experimenter can stop sampling early to influence the results. It emphasizes the importance of robust statistical methods to guard against the effects of optional stopping and the need for empirical studies to approximate the operating characteristics of sequential designs. Overall, the chapter underscores the potential of sequential design in enhancing statistical efficiency and the ongoing efforts to develop more sophisticated methods in this area.