In the first part of this paper a sequential detection procedure, based on a Sequential Sample Median Test (S.S.M.T.), is introduced for the problem of discriminating between shift alternatives (constant signals in additive noise), in a specified noise environment. This detection procedure is shown to have the desirable properties of ease of implementation and of comparing favorably with efficient rank vector detection procedures. The decision samples are acquired in data groups of M samples on which an intermediate decision, based on the value of the sample median of the data group, is made. The final decision to accept one of the two hypotheses is based on the sum of the intermediate decisions. Each intermediate decision is made by comparing the sample median to two decision thresholds. These thresholds are determined by (i) having the probability of error and average sample size under the two hypotheses equal and (ii) specifying either the average sample size or average probability of error. For both types of constraints a numerical optimization procedure to find the “optimum≓ value of M is used. Specific results are obtained for additive gaussian and Cauchy noise. However, to apply this test to either problem it is necessary to know the c.d.f. of the decision samples, i.e., the decision thresholds are a function of the additive noise distribution, and this c.d.f. may not be known. In the second part of this paper an extension of the S.S.M.T. to an unspecified noise environment, for the average sample size constraint, is presented. Two recursive procedures are developed for estimating the decision thresholds corresponding to the desired average sample size. Both procedures are based on the Robbins-Monro stochastic approximation concept and are shown to converge with probability 1 to the correct decision threshold.