Optimal bounding ellipsoid (OBE) algorithms comprise a class of novel recursive identification methods for affine-in-parameters system and signal models. OBE algorithms exploit a priori knowledge of bounds on model disturbances to derive a monotonically nonincreasing set of solutions that are feasible in light of the observations and the model structure. In turn, these sets admit criteria for ascertaining the information content of incoming observations, obviating the expense of updating when data are redundant. Relative to classical recursive methods for this task, OBE algorithms are efficient, robust, and exhibit remarkable tracking ability, rendering them particularly attractive for realtime signal processing and control applications. After placing the OBE algorithms in the hierarchy of the broader set-membership identification methods, this article introduces the underlying set-theoretic concepts, compares and contrasts the various published OBE algorithms including the motivation for each development, then concludes with some illustrations of OBE algorithm performance. More recent work on the use of OBE processing infiltering tasks is also included in the discussion. The paper is a survey of a broad and evolving topic, and extensive references to further information are included.