Superplasticity is the phenomenon of extraordinary ductility exhibited by some alloys with extremely fine grain size, when deformed at elevated temperatures and in certain ranges of strain rate. To put the phenomenology on a proper basis, careful mechanical tests are necessary. These are divided into (i) primary creep tests, (ii) steady state deformation tests, and (iii) instability and fracture tests, all of which lead to identification of macroscopic parameters. At the same time, microstructural observations establish those characteristics that are pre-requisites for superplastic behaviour. Among the macroscopic characteristics to be explained by any theory is a proper form of the equation for the strain rate as a function of stress, grain size and temperature. It is commonly observed that the relationship between stress and strain rate at any temperature is a continuous one that has three distinct regions. The second region covers superplastic behaviour, and therefore receives maximum attention. Any satisfactory theory must also arrive at the dependence of the superplastic behaviour on the various microstructural characteristics. Theories presented so far for microstructural characteristics may be divided into two classes: (i) those that attempt to describe the macroscopic behaviour, and (ii) those that give atomic mechanisms for the processes leading to observable parameters. The former sometimes incorporate micromechanisms. The latter are broadly divided into those making use of dislocation creep, diffusional flow, grain boundary deformation and multimechanisms. The theories agree on the correct values of several parameters, but in matters that are of vital importance such as interphase grain boundary sliding or dislocation activity, there is violent disagreement. The various models are outlined bringing out their merits and faults. Work that must be done in the future is indicated.