Serializability has been widely accepted as the correctness criterion for databases subject to concurrent access. However, in a number of the newer and most challenging application areas, serializable execution may not be feasible. A number of models have recently been proposed in which transactions are decomposed into smaller, atomic, interleavable steps. These models have the potential for improving performance since locks are released at the end of each step. Models of distributed transactions have also been proposed with the similarity that subtransactions correspond to steps. In most of this work serializability is no longer guaranteed. In this paper we propose a new, application-oriented, correctness criterion that utilizes transaction semantics. We treat transactions as programs whose semantics can be analyzed at design time. The effect of each transaction is specified using pre- and postconditions, and any schedule that preserves these conditions is permissible. Such schedules can produce database states that could not be reached by any serial execution. In addressing the issue of performance, we use transaction semantics to decompose transactions into steps and describe a concurrency control that controls step interleaving in such a way that assertions are preserved. The same model can be used to study the interleaving of subtransactions of concurrent distributed transactions.