Designs for sequential sampling procedures that adapt to cumulative information are discussed. A familiar illustration is the play-the-winner rule in which there are two treatments; after a random start, the same treatment is continued as long as each successive subject registers a success. When a failure occurs, the other treatment is used until failure occurs. Sequential sampling procedures are discussed in relation to clinical trials, but there are many applications of such procedures in statistics. Models are presented for the adaptive allocation of dose levels to subjects arriving sequentially. Natural multivariate design spaces occurring in radiotherapy given at the Fred Hutchinson Cancer Research Center motivate this work. The allocation rules proposed adapt to accumulating failure data, possibly censored, and thus are governed by an underlying, unknown, multivariate failure distribution in the limit. An urn model is used to effect the adaptation of the allocation scheme. A recursive algorithm for updating the urn distribution is derived. The Bernoulli environment and urn models on ordinal sample spaces in a random environment are also considered. (SLD)