Generalized sequential probability ratio tests (hereafter abbreviated GSPRT's) for testing between two simple hypotheses have been defined in [1]. The present paper, divided into four sections, discusses certain properties of GSPRT's. In Section 1 it is shown that under certain conditions the distributions of the sample size under the two hypotheses uniquely determine a GSPRT. In the second section, the admissibility of GSPRT's is discussed, admissibility being defined in terms of the probabilities of the two types of error and the distributions of the sample size required to come to a decision; in particular, notwithstanding the result of Section 1, many GSPRT's are inadmissible. In Section 3 it is shown that, under certain monotonicity assumptions on the probability ratios, the GSPRT's are a complete class with respect to the probabilities of the two types of error and the average distribution of the sample size over a finite set of other distributions. In Section 4, finer characterizations are given of GSPRT's which minimize the expected sample size under a third distribution satisfying certain monotonicity properties relative to the other two distributions; these characterizations give monotonicity properties of the decision bounds.