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We study the problem of generating synthetic databases having declaratively specified characteristics. This problem is motivated by database system and application testing, data masking, and benchmarking. While the data generation problem has been studied before, prior approaches are either non-declarative or have fundamental limitations relating to data characteristics that they can capture and efficiently support. We argue that a natural, expressive, and declarative mechanism for specifying data characteristics is through cardinality constraints; a cardinality constraint specifies that the output of a query over the generated database have a certain cardinality. While the data generation problem is intractable in general, we present efficient algorithms that can handle a large and useful class of constraints. We include a thorough empirical evaluation illustrating that our algorithms handle complex constraints, scale well as the number of constraints increase, and outperform applicable prior techniques.