Heat exchanger network (HEN) design problem has attracted great attention in scientific literature. A large number of papers were published to date including several reviews, the last one in 1988. There are also successful industrial applications of HEN design approaches. The main aim of this review is to show an evolution of HEN design approaches which have produced the currently ＇＇leading＇＇ subroutines. The review paper is divided into two parts. The first one addresses the problem of heat exchanger grassroot design, i.e., the synthesis of HEN-s. More specifically this part deals with targeting approaches and synthesis methods based on deep insight into the problem. These synthesis methods are deeply connected with HEN targeting concepts and techniques. The second part of the review presents mathematical/optimization based approaches for HEN synthesis and addresses the problem of HEN retrofit design. In the both parts the main impact is put on the more recent developments, from about 1988 albeit the basic concepts presented before 1988 are also described since they are necessary for understanding more recent methodologies.