Most materials demonstrate an expansion upon heating, however a few are known to contract, i.e. exhibit a negative coefficient of thermal expansivity (NTE). This naturally occurring phenomenon has been shown to occur in a range of solids including complex metal oxides, polymers and zeolites, and opens the door to composites with a coefficient of thermal expansion (CTE) of zero. The state of the art in NTE solids is reviewed, and understanding of the driving mechanisms of the effect is considered along with experimental and theoretical evidence. The various categories of solids with NTE are explored, and experimental methods for their experimental characterisation and applications for such solids are proposed. An abstraction for an underlying mechanism for NTE at the supramolecular level and its applicability at the molecular level is discussed.