This is a comprehensive discussion of complexity as it arises in physical, chemical, and biological systems, as well as in mathematical models of nature. Common features of these apparently unrelated fields are emphasised and incorporated into a uniform mathematical description, with the support of a large number of detailed examples and illustrations. All necessary definitions have been given in as rigorous a way as allowed, without sacrificing clarity and readability. The uniformity of notation and the relegation of technical mathematical notions to the appendices will help the reader follow the main course of the discussion without much need for consulting standard textbooks or the original research papers.