In this paper, a survey on systems which exhibit gains and/or dynamics depending on the input or output direction is presented. Several examples of such systems covering a wide range of application areas, and the causes of their direction-dependent responses, are given. Direction-dependent systems are mathematically classified into three classes, depending on their state-space representation and switching function. The related linear dynamic, transient, and steady-state characteristics of these systems are investigated. Approaches to model direction-dependent systems using Wiener models, piecewise linear models, bilinear models and recurrent neural networks are described. Three application examples are used to illustrate the usefulness of the direction-dependent description. The advantages and disadvantages of direction-dependent models, as well as possible directions for future work, are pointed out. (C) 2009 Elsevier Ltd. All rights reserved.