  Power laws and distributions with heavy tails are common features of many
experimentally studied complex systems, like the distribution of the sizes of
earthquakes and solar flares, or the duration of neuronal avalanches in the
brain. Previously, researchers surmised that a single general concept may act
as a unifying underlying generative mechanism, with the theory of self
organized criticality being a weighty contender.
  Consequently, a substantial amount of effort has gone into developing new and
extended models and, hitherto, three classes of models have emerged. The first
line of models is based on a separation between the time scales of drive and
dissipation, and includes the original sandpile model and its extensions, like
the dissipative earthquake model. Within this approach the steady state is
close to criticality in terms of an absorbing phase transition. The second line
of models is based on external drives and internal dynamics competing on
similar time scales and includes the coherent noise model, which has a
non-critical steady state characterized by heavy-tailed distributions. The
third line of models proposes a non-critical self-organizing state, being
guided by an optimization principle, such as the concept of highly optimized
tolerance.
  We present a comparative overview regarding distinct modeling approaches
together with a discussion of their potential relevance as underlying
generative models for real-world phenomena. The complexity of physical and
biological scaling phenomena has been found to transcend the explanatory power
of individual paradigmal concepts. The interaction between theoretical
development and experimental observations has been very fruitful, leading to a
series of novel concepts and insights.
