Power grids, the human brain, arrays of coupled lasers, genetic
networks, or the Amazon rainforest are all characterized by
multistability. For power grids, the strongly ongoing transition to
distributed renewable energy sources leads to a proliferation of
dynamical actors. The desynchronization of a few or even one of those
would likely result in a substantial blackout. Thus, the dynamical
stability of the synchronous state has become a leading topic in power
grid research.  The likelihood that such systems will remain in the
most desirable of their many stable states depends on their stability
against significant perturbations, particularly in a state space
populated by undesirable states. Here we claim that the traditional
linearization-based approach to stability is in several cases too
local to adequately assess how stable a state is. Instead, we quantify
it in terms of basin stability, a new measure related to the volume of
the basin of attraction. Basin stability is non-local, nonlinear and
easily applicable, even to high-dimensional systems. It provides a
long-sought-after explanation for the surprisingly regular topologies
of neural networks and power grids, which have eluded theoretical
description based solely on linear stability.  Remarkably, when taking
physical losses in the network into account, the back-reaction of the
network induces new exotic solitary states in the individual actors,
and the stability characteristics of the synchronous state are
dramatically altered. These novel effects will have to be explicitly
taken into account in the design of future power grids, and their
existence poses a challenge for control.