Abstract Concepts used in the scientific study of complex systems have become so widespread that their use and abuse has led to ambiguity and confusion in their meaning. In this paper we use information theory to provide abstract and concise measures of complexity, emergence, self-organization, and homeostasis. The purpose is to clarify the meaning of these concepts with the aid of the proposed formal measures. In a simplified version of the measures (focusing on the information produced by a system), emergence becomes the opposite of self-organization, while complexity represents their balance. Homeostasis can be seen as a measure of the stability of the system. We use computational experiments on random Boolean networks and elementary cellular automata to illustrate our measures at multiple scales. Nontechnical Abstract There several measures and definitions of complexity, emergence, self-organization, and homeostasis. This has led to confusion reflected on the carefree use of these concepts. We provide precise definitions based on information theory, attempting to clarify the meaning of these concepts with simple but formal measures. We illustrate the measures with experiments on discrete dynamical systems. Keywords : complexity, information, emergence, self-organization, homeostasis. 1 Introduction In recent decades, the scientific study of complex systems has increased our understanding of a broad range of phenomena and produced several useful tools (Bar-Yam, 1997; Mitchell, 2009). Concepts used to characterize complex systems—such as emergence, adaptivity, self- organization, and complexity itself—have been used in different contexts with different mean- ings (Nicolis and Prigogine, 1977; Haken, 1988; Holland, 1995; Schweitzer, 1997; Wolfram, 2