The aim of the paper is to review critically the notion of autopoiesis as presented by Maturana and Varela. In particular, recognizing that there are difficulties in obtaining a complete and clear picture from the primary literature, an effort is made to present a coherent view-also based on many years of personal contact with Francisco Varela. The paper begins with a few historical notes to highlight the cultural background from which the notion of autopoiesis arose. The basic principles of autopoiesis as a theory of cellular life are then described, emphasizing also what autopoiesis is not: not an abstract theory, not a concept of artificial life, not a theory about the origin of life-but rather a pragmatic blueprint of life based on cellular life. It shown how this view leads to a conceptually clear definition of minimal life and to a logical link with related notions, such as self-organization, emergence, biological autonomy, auto-referentiality, and interactions with the environment. The perturbations brought about by the environment are seen as changes selected and triggered by the inner organization of the living. These selective coupling interactions impart meaning to the minimal life and are thus defined by Maturana and Varela with the arguable term of "cognition". This particular view on the mutual interactions between living organism and environment leads these authors to the notion of "enaction", and to the surprising view that autopoiesis and cognition are two complementary, and in a way equivalent, aspects of life. It is then shown how cognition, so defined, permits us to build a bridge between biology and cognitive science. Autopoiesis also allows one to conceive chemical models of minimal cellular life that can be implemented experimentally. The corresponding work on "chemical autopoiesis" is then reviewed. The surprising impact of autopoiesis in the social sciences ("social autopoiesis") is also briefly discussed. This review also comments on why the theory of autopoiesis had, and still has, a difficult time being accepted into the mainstream of life-science research. Finally, it is pointed out that the new interest in system biology and complexity theories may lead to a reappraisal of autopoiesis and related notions, as outlined also by other authors, such as Tibor Ganti and Stuart Kauffmann.