This review presents the development of brush copolymers as macromolecular engineering materials prepared by atom transfer radical polymerization (ATRP) and reported by the Matyjaszewski's group during the last twenty years. The graft structures combine different segments which allowed to precisely design properties of macromolecules to target various applications. The first molecular brushes contained homopolymer side chains, which were extended by other segments yielding core-shell macromolecules with block copolymer side chains. Many variants of brushes, densely vs loosely grafted copolymers, statistical and gradient distribution of grafts, block and statistical graft structures, homografted vs heterografted copolymers, have been designed. The semigrafted copolymers combine linear and brush topologies, whereas block brushes consist of two different grafted segments. These are also known the brushes with much more complex architectures, such as star-like brushes. Examples of various types of brushes, properties and their potential applications are discussed. (C) 2015 Elsevier Ltd. All rights reserved.