  While all materials reduce their intrinsic volume under hydrostatic (uniform)
compression, a select few actually \emph{expand} along one or more directions
during this process of densification. As rare as it is counterintuitive, such
"negative compressibility" behaviour has application in the design of pressure
sensors, artificial muscles and actuators. The recent discovery of surprisingly
strong and persistent negative compressibility effects in a variety of new
families of materials has ignited the field. Here we review the phenomenology
of negative compressibility in this context of materials diversity, placing
particular emphasis on the common structural motifs that recur amongst known
examples. Our goal is to present a mechanistic understanding of negative
compressibility that will help inform a clear strategy for future materials
design.
