MAPPINGS III is a general purpose astrophysical plasma modelling code. It is principally intended to predict emission line spectra of medium and low density plasmas subjected to different levels of photoionization and ionization by shockwaves. MAPPINGS III tracks up to 16 atomic species in all stages of ionization, over a useful range of 102 to 108 K. It treats spherical and plane parallel geometries in equilibrium and time-dependent models. MAPPINGS III is useful for computing models of HI and HII regions, planetary nebulae, novae, supernova remnants, Herbig-Haro shocks, active galaxies, the intergalactic medium and the interstellar medium in general. The present version of MAPPINGS III is a large FORTRAN program that runs with a simple TTY interface for historical and portability reasons. A newer version of this software, MAPPINGS V (ascl:1807.005), is available.