  In this chapter of the e-book "Self-Organized Criticality Systems" we
summarize some theoretical approaches to self-organized criticality (SOC)
phenomena that involve percolation as an essential key ingredient. Scaling
arguments, random walk models, linear-response theory, and fractional kinetic
equations of the diffusion and relaxation type are presented on an equal
footing with theoretical approaches of greater sophistication, such as the
formalism of discrete Anderson nonlinear Schr\"odinger equation, Hamiltonian
pseudochaos, conformal maps, and fractional derivative equations of the
nonlinear Schr\"odinger and Ginzburg-Landau type. Several physical consequences
are described which are relevant to transport processes in complex systems. It
is shown that a state of self-organized criticality may be unstable against a
bursting ("fishbone") mode when certain conditions are met. Finally we discuss
SOC-associated phenomena, such as: self-organized turbulence in the Earth's
magnetotail (in terms of the "Sakura" model), phase transitions in SOC systems,
mixed SOC-coherent behavior, and periodic and auto-oscillatory patterns of
behavior. Applications of the above pertain to phenomena of magnetospheric
substorm, market crashes, and the global climate change and are also discussed
in some detail. Finally we address the frontiers in the field in association
with the emerging projects in fusion research and space exploration.

Chaotic Dynamics;Disordered Systems and Neural Networks;Adaptation and Self-Organizing Systems