  This mini-review synthesizes our understanding of the equilibrium behavior of
particle models with short-range attractive and long-range repulsive (SALR)
interactions. These models, which can form stable periodic microphases, aim to
reproduce the essence of colloidal suspensions with competing interactions.
Ordered structures, however, have yet to be obtained in experiments. In order
to better understand the hurdles to periodic microphase assembly, marked
theoretical and simulation advances have been made over the last few years.
Here, we present recent progress in the study of microphases in models with
SALR interactions using liquid-state theory and density-functional theory as
well as numerical simulations. Combining these various approaches provides a
description of periodic microphases, and give insights into the rich
phenomenology of the surrounding disordered regime. Three additional ongoing
research directions in the thermodynamics of models with SALR interactions are
also presented.
