  Localized patterns are coherent structures embedded in a quiescent state and
occur in both discrete and continuous media across a wide range of
applications. While it is well-understood how domain covering patterns (for
example stripes and hexagons) emerge from a pattern-forming/Turing instability,
analyzing the emergence of their localized counterparts remains a significant
challenge. There has been considerable progress in studying localized patterns
over the past few decades, often by employing innovative mathematical tools and
techniques. In particular, the study of localized pattern formation has
benefited greatly from numerical techniques; the continuing advancement in
computational power has helped to both identify new types patterns and further
our understanding of their behavior. We review recent advances regarding the
complex behavior of localized patterns and the mathematical tools that have
been developed to understand them, covering various topics from spatial
dynamics, exponential asymptotics, and numerical methods. We observe that the
mathematical understanding of localized patterns decreases as the spatial
dimension increases, thus providing significant open problems that will form
the basis for future investigations.
