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We extend the standard for object-oriented databases, ODMG, with reactive features, by proposing a language for specifying triggers and defining its semantics. This extension has several implications, thus we make three different specific contributions. First, the definition of a declarative data manipulation language for ODMG, which is missing in the current version of the standard; such a definition requires revisiting data manipulation in ODMG and also addressing issues related to set-oriented versus instance-oriented computation. Then, the definition of a trigger language for ODMG, unifying also the SQL:1999 proposal and providing support for trigger inheritance and overriding. Finally, the development of a formal semantics for the proposed data manipulation and trigger languages.