Updates are a crucial component of any database programming language. Even the simplest database transactions, such as withdrawal from a bank account, require updates. Unfortunately, updates are not accounted for by the classical Horn semantics of logic programs and deductive databases, which limits their usefulness in real-world applications. As a short-term practical solution, logic programming languages have resorted to handling updates using ad hoc operators without a logical semantics. A great many works have been dedicated to developing logical theories in which the state of the underlying database can evolve with time. Many of these theories were developed with specific applications in mind, such as reasoning about actions, database transactions, program verification, etc. As a result, the different approaches have different strengths and weaknesses. In this survey, we review a number of these works, discuss their application domains, and highlight their strong and weak points.