.We study properties of rewrite systems that are not necessarily terminating, but allow instead for trans#nite derivations that have a limit. In particular, we give conditions for the existence of a limit and for its uniqueness and relate the operational and algebraic semantics of in#nitary theories. We also consider su#cient completeness of hierarchical systems. Is there no limit? ---Job 16:3 1. Introduction Rewrite systems are sets of directed equations used to compute by repeatedly replacing equal terms in a given formula, as long as possible. For one approach to their use in computing, see #23#. The theory of rewriting is an outgrowth of the study of the lambda calculus and combinatory logic, and # Preliminary versions #6, 7# of ideas in this paper were presented at the Sixteenth ACM Symposium on Principles of Programming Languages, Austin, TX #January 1989# and at the Sixteenth EATCS International Colloquium on Automata, Languages and Programming, Stresa, Italy #July 1989#. ...