  A major challenge of interdisciplinary description of complex system
behaviour is whether real systems of higher complexity levels can be understood
with at least the same degree of objective, "scientific" rigour and
universality as "simple" systems of classical, Newtonian science paradigm. The
problem is reduced to that of arbitrary, many-body interaction (unsolved in
standard theory). Here we review its causally complete solution, the ensuing
concept of complexity and applications. The discovered key properties of
dynamic multivaluedness and entanglement give rise to a qualitatively new kind
of mathematical structure providing the exact version of real system behaviour.
The extended mathematics of complexity contains the truly universal definition
of dynamic complexity, randomness (chaoticity), classification of all possible
dynamic regimes, and the unifying principle of any system dynamics and
evolution, the universal symmetry of complexity. Every real system has a
non-zero (and actually high) value of unreduced dynamic complexity determining,
in particular, "mysterious" behaviour of quantum systems and relativistic
effects causally explained now as unified manifestations of complex interaction
dynamics. The observed differences between various systems are due to different
regimes and levels of their unreduced dynamic complexity. We outline
applications of universal concept of dynamic complexity emphasising cases of
"truly complex" systems from higher complexity levels (ecological and living
systems, brain operation, intelligence and consciousness, autonomic information
and communication systems) and show that the urgently needed progress in social
and intellectual structure of civilisation inevitably involves qualitative
transition to unreduced complexity understanding (we call it "revolution of
complexity").
