|Artist’s interpretation of complexity, of a lecture by Paul Davies entitled “Taming complexity: opening a new scientific frontier,” delivered at the Future Summit of the Australian Davos Connection in Sydney, Australia 12 May 2008).|
The term complexity theory or complexity science, all-in-all, is a general umbrella term for the study of a number of disperse fields that use mathematical methods of investigation, without any uniform framework or core principles; an example being the use of Edwin Jaynes' so-called maximum entropy principle, blended together with Riemannian metrics to explain complex systems. 
The so-called science of complexity tends to refer to an admixture of 1990s going into the 2000s theories, for the most part chaos theory mixed together with computer simulations.
The following, for example, is a 2009 complexity theory-science flow chart style timeline is American medical sociologist Brian Castellani’s interpretation of complexity science, which shows, rather correctly, that what is often referred to as “complexity theory” (or complexity science), essentially, is an umbrella term for a mixture of a number of related cousin subjects, in particular Austrian biologist Ludwig Bertalanffy’s 1940 general systems theory and American mathematician Norbert Wiener’s 1948 cybernetics theory, in core structure, with appendage additions, including: Kenneth Boulding (economic systems), Howard T. Odum (ecological systems), James Miller (biological systems theory), James Lovelock and Lynn Margulis (gaia hypothesis), John Neumann (mathematics), Claude Shannon (information theory), Edgar Morin (sociology and philosophy), Fritjof Capra (web of life), Niklas Luhmann (sociology), Erich Jantsch (self-organizing universe), Ilya Prigogine (dissipative structures), Stuart Kauffman (autocatalytic emergence), Albert-Lazlo Barabasi (linked theory), Mark Granovetter (weak ties), all connected together with a general kindred towards computer modeling and chaos theory. 
It is difficult to say, however, as of 2010, if complexity is a legitimate branch of science. In any event, in sum, complexity theory, in biology and sociology, is often used as an umbrella term for a mixture of chaos theory, cybernetics, information theory, general systems theory, along with superficial mentions of thermodynamics, among others.
Belgian chemist Ilya Prigogine dates the start of complexity science to the heat transfer work of Fourier: 
“As for the birth of the ‘science of complexity’, we propose to date it in 1811, the year Joseph Fourier won the prize of the French Academy of Science for his mathematical description of the propagation of heat in solids.”
This, however, is a biased misattribution, being that Fourier's work forms the basis of the subject of heat transfer
1. Bothamley, Jennifer. (2002). Dictionary of Theories: One Stop to more than 5,000 Theories. Visible Ink.
2. Prigogine, Ilya. (1984). Order Out of Chaos – Man’s New Dialogue with Nature (pg. 104). New York: Bantam Books.
3. Dewar, Robert L. Detering, Frank. (2009). Complex Physical, Biophysical, and Econophysical Systems: Proceedings of the 22nd Canberra International Physics Summer School. World Scientific.
4. (a) Complexity science map – Art-ScienceFactory.com.
(b) Castellani, Brian and Hafferty, Frederic. (2009). Sociology and Complexity Science. Springer.
● Gregoire, Nicolis and Prigogine, Illya. (1989). Exploring Complexity - an Introduction. New York: Freeman and Co.
● Complexity – Wikipedia.