The concept of complexity has become very important in theoretical biology. It is a many faceted concept and too new and ill defined to have a universally accepted meaning. This review examines the development of this concept from the point of view of its usefulness as a criteria for the study of living systems to see what it has to offer as a new approach. In particular, one definition of complexity has been put forth which has the necessary precision and (...) rigor to be considered as a useful categorization of systems, especially as it pertains to those we call living. This definition, due to Robert Rosen, has been developed in a number of works and involves some deep new concepts about the way we view systems. In particular, it focuses on the way we view the world and actually practice science through the use of the modelling relation. This mathematical object models the process by which we assign meaning to the world we perceive. By using the modelling relation, it is possible to identify the subjective nature of our practices and deal with this issue explicitly. By so doing, it becomes clear that our notion of complexity and especially its most popular manifestations, is in large part a product of the historical processes which lead to the present state of scientific epistemology. In particular, it is a reaction to the reductionist/mechanistic view of nature which can be termed the Newtonian Paradigm. This approach to epistemology has dominated for so long that its use as a model has become implicit in most of what we do in and out of science. The alternative to this approach is examined and related to the special definition of complexity given by Rosen. Some historical examples are used to emphasize the dependence of our view of what is complex in a popular sense on the ever changing state of our knowledge. The role of some popular concepts such as chaotic dynamics are examined in this context. The fields of artificial life and related areas are also viewed from the perspective of this rigorous view of complexity and found lacking. The notion that in some way life exists at the edge of chaos is examined from the perspective of the second law of thermodynamics given by Schneider and Kay. Finally, the causal elements in complex systems are explored in relation to complexity. Rosen has shown that a clear difference in causal relations exists between complex and simple systems and that this difference leads to a uniquely useful definition of what we mean by living. Rosen makes it very clear that the class of systems which are complex is a much larger class than those which we call living. For that reason, the focus of this review will be on complexity as a stepping stone towards the deeper question of what makes a system alive. (shrink)