  The interplay between short-range attractions and long-range repulsions
(SALR) characterizes the so called liquids with competing interactions, which
are known to exhibit a variety of equilibrium and non-equilibrium phases. The
theoretical description of the phenomenology associated to glassy or gel states
in these systems has to take into account both, the presence of thermodynamic
instabilities (such as those defining the spinodal line and the so called
$\lambda$ line) and the limited capability to describe genuine non-equilibrium
processes from first principles. Here we report the first application of the
non-equilibrium self-consistent generalized Langevin equation theory, to the
description of the dynamical arrest processes that occur in SALR systems after
being instantaneously quenched into a state point in the regions of
thermodynamic instability. The physical scenario predicted by this theory
reveals an amazing interplay between the thermodynamically-driven
instabilities, favoring equilibrium macro- and micro-phase separation, and the
kinetic arrest mechanisms, favoring non-equilibrium amorphous solidification of
the liquid into an unexpected variety of glass and gel states.
