Abstract. A review is presented of the theory of pattern forma- tion in extended, dissipative, highly nonequilibrium systems. Emphasis is placed on systems which, in addition to spatial translations and rotations, have additional continuous symme- try group(s). It is shown that the additional symmetry destabi- lizes dramatically the ground state of the system thus causing it to make a direct transition from a spatially uniform to a turbu- lent state in an analogous fashion to the second-order phase transition in quasi-equilibrium systems. Apart from the theore- tical analysis, a discussion of experimental data is given. To the respectful memory of I M Lifshitz. 1. Introduction I regard it as a great honour to have my paper published in the special issue of Physics Uspekhi dedicated to I M Lifshitz on the memorial of his 80th birthday. It is a great responsibility too, bearing in mind the problems with which the present review is concerned. In fact, the field the review belongs to, namely self-organization and transition to chaos, attracted much of I M Lifshitz' attention during the last years of his life. I personally believe that in spite of the fact that he has not published any paper devoted to the subject, this was mainly due to the fact that he had been collecting information in this field that was new to him. This preliminary work would most certainly have been crowned with an article (or a series of papers) laying down a programme of further progress, as had been the case of his studies in polymer physics [1]. It would have been... but it did not take place Ð this hope was dashed by the premature death of I M Lifshitz. The objective of the present review is to draw attention of research scientists to selected results of recent studies on pattern formation and transition to chaos in dissipative systems. These results in many respects digress from tradi- tional concepts in this field indicating that the symmetry of the problem is of greater importance than it was believed to be till now. The review is addressed to a broad audience of both specialists and non-experts. For this reason I shall try to avoid, whenever appropriate, cumbersome mathematical calculations, which are normally inseparable from the theo- retical description of the problems and refer the readers to original papers for details. This is supposed to allow to focus on the key aspects of the analysis with special reference to specific qualitative features of the phenomena of interest. Terminology relevant to the problems discussed is this review has not settled yet, and some authors read their own thoughts into the terms which actually have quite different sense. To avoid misunderstanding, I shall start with explain- ing the meaning of special terms which the reader will encounter below. Distributed systems are any objects or phenomena described by equations in partial derivatives (mainly, non- linear ones). The term dissipative implies that the considered systems are non-equilibrium and certain dissipative processes occur. In the case of open systems (where energy is fed from the outside to be further dissipated and eventually removed from the system or released `to infinity'), `the energy flux through the system' is supposed to remain constant. This M I Tribel'ski|¯ Graduate School of Mathematical Science, University of Tokyo 3-8-1 Komaba, Mediro-ku Tokyo 153, Japan E-mail: tribel@ms.u-tokyo.ac.jp Received 22 October 1996 Uspekhi Fizicheskikh Nauk 167 (2) 167 ± 190 (1997) Translated by Yu V Morozov; edited by A I Yaremchuk REVIEWS OF TOPICAL PROBLEMS PACS numbers: 05.45.+b, 47.52.+j Short-wavelength instability and transition to chaos in distributed systems with additional symmetry M I Tribel'ski |¯ { Contents 1. Introduction 159 2. Short-wavelength instability in conventional systems 162 2.1 Universality of the Swift ± Hohenberg and Ginzburg ± Landau equations; 2.2 Spatially periodic solutions and their stability. Eckhaus criterion 3. Short-wavelength instability in systems with an additional continuous group of symmetry 167 3.1 Symmetry and Goldstone modes; 3.2 (Quasi)one-dimensional spatially periodic solutions and their stability; 3.3 Symmetry and dispersion equation in case of mixed e-scales 4. Soft turbulent modes and `continuous' transition to chaos 172 5. Experiment. Turbulence with zero critical Reynolds number 174 6. Conclusion 177 7. Appendix 178 References 179 Physics ± Uspekhi 40 (2) 159 ± 180 (1997) #1997 Uspekhi Fizicheskikh Nauk, Russian Academy of Sciences { The author is also known as M I Tribelsky. The name used here is a transliteration under the BSI/ANSI scheme adopted by this journal.