In this article, we consolidate several literature reports in an attempt to provide a structural definition of a polymer brush as a tethered polymer layer with a high grafting density of polymer chains. We will try to distinguish a brush from grafted polymer layers and coated layers and provide a reasonable approach to the description of the brush regime with a single parameter Sigma - reduced grafting density. Furthermore, we will attempt to describe the relevance of this description to applications. This article is timely given the continuing interest relating to brushes on flat substrates, nanoparticles, and the walls of micro-channels. (C) 2007 Wiley Periodicals, Inc.