We study active integrity constraints and revision programming, two formalisms designed to describe integrity constraints on databases and to specify policies on preferred ways to enforce them. Unlike other more commonly accepted approaches, these two formalisms attempt to provide a declarative solution to the problem. However, the original semantics of founded repairs for active integrity constraints and justified revisions for revision programs differ. Our main goal is to establish a comprehensive framework of semantics for active integrity constraints, to find a parallel framework for revision programs, and to relate the two. By doing so, we demonstrate that the two formalisms proposed independently of each other and based on different intuitions when viewed within a broader semantic framework turn out to be notational variants of each other. That lends support to the adequacy of the semantics we develop for each of the formalisms as the foundation for a declarative approach to the problem of database update and repair. In the paper we also study computational properties of the semantics we consider and establish results concerned with the concept of the minimality of change and the invariance under the shifting transformation.