Databases are often incomplete because of the presence of disjunctive information, due to con icts, partial knowledge and other reasons. Queries against such databases often ask questions about various possibilities encoded by the stored data, rather than the stored data itself. Normalization, which is a mechanism for asking such queries, was presented in LW93a]; however, it had exponential space complexity. The main goal of this paper is to develop a general theory of answering queries against incomplete databases with disjunctive information, and use it to design practical algorithms for query evaluation. We de ne the semantics of such databases and prove normalization theorems for set- and bag-based complex objects. These theorems provide us with programming primitives that one needs in order to obtain the list of all possibilities encoded by a complex object with disjunctions. We study two ways of making query evaluation faster and more space e cient. Partial normalization allows ...