Sequential sampling schemes have traditionally used ad hoc rules for sample size. The variable-sample-size sequential probability ratio test (VPRT), developed by Cressie and Morgan (1986), generalizes the Wald-Wolfowitz one-at-a-time sampling plan. The VPRT finds the sample size that maximizes the expected net gain of sampling, conditional on the accumulated data at hand. In this dissertation, we explore this idea, both theoretically and numerically, using a technique of computation known as backward induction. Applications to sequential testing of means of Gaussian data, and to a sequential procedure of testing efficacy of a pharmaceutical (developed by Berry and Ho, 1988) are presented. The relevance of sampling size optimization to sequential clinical trials is discussed;References. (1) Berry, D. A. and Ho, C-H. 1988. One-Sided Sequential Stopping Boundaries for Clinical Trials: A Decision-Theoretic Approach. Biometrics, 44: 219-227. (2) Cressie, N., and Morgan, P. B. 1986. The VPRT: A Sequential Testing Procedure Dominating the SPRT. Preprint No. 86-17. Statistical Laboratory, Iowa State University, Ames.