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Transactional memory is perceived as an appealing alternative to critical sections for general purpose concurrent programming. Despite the large amount of recent work on transactional memory implementations, however, its actual specification has never been precisely defined. This paper presents \emph{opacity}, a new correctness criterion for transactional memory systems. Opacity extends the notion of strict serializability, itself a strong form of the classical serializability property, with the requirement that even \emph{non-committed} transactions are prevented from accessing inconsistent state. Yet opacity does not preclude versioning, invisible reads and lazy updates, often used by modern TM implementations. In fact, most transactional memory systems we know of ensure opacity. We prove a tight bound on the inherent cost of implementing opacity. The bound highlights a trade-off that explains some of the differences between current transactional memory systems, and also draws a sharp complexity line between opacity on one hand, and the combination of strict serializability and strict recoverability on the other hand.