  Scale-free outbursts of activity are commonly observed in physical,
geological, and biological systems. The idea of self-organized criticality
(SOC), introduced back in 1987 by Bak, Tang and Wiesenfeld suggests that, under
certain circumstances, natural systems can seemingly self-tune to a critical
state with its concomitant power-laws and scaling. Theoretical progress allowed
for a rationalization of how SOC works by relating its critical properties to
those of a standard non-equilibrium second-order phase transition that
separates an active state in which dynamical activity reverberates
indefinitely, from an absorbing or quiescent state where activity eventually
ceases. Here, we briefly review these ideas as well as a recent closely-related
concept: self-organized bistability (SOB). In SOB, the very same type of
feedback operates in a system characterized by a discontinuos phase transition,
which has no critical point but instead presents bistability between active and
quiescent states. SOB also leads to scale-invariant avalanches of activity but,
in this case, with a different type of scaling and coexisting with anomalously
large outbursts. Moreover, SOB explains experiments with real sandpiles more
closely than SOC. We review similarities and differences between SOC and SOB by
presenting and analyzing them under a common theoretical framework, covering
recent results as well as possible future developments. We also discuss other
related concepts for "imperfect" self-organization such as "self-organized
quasi-criticality" and "self-organized collective oscillations", of relevance
in e.g. neuroscience, with the aim of providing an overview of feedback
mechanisms for self-organization to the edge of a phase transition.
