We present new broadband optical and near-infrared CCD imaging together with deep optical intermediate-resolution spectroscopy of the Hickson Compact Group 31. We analyze the morphology and colors of the stellar populations of the galaxies, as well as the kinematics, physical conditions, and chemical composition of the ionized gas in order to get a more complete view on the origin and evolution of the system. We estimate the ages of the most recent star formation bursts of the system, finding an excellent consistency among the values obtained with different indicators and starburst models. We find that member F hosts the youngest starburst of the group, showing a substantial population of Wolf-Rayet stars. The chemical abundances are fairly similar in all the members of the group despite their very different absolute magnitudes. We argue that the use of traditional metallicity-luminosity relations based on the absolute B-magnitude is not appropriate for dwarf starburst galaxies, because their luminosity is dominated by the transient contribution of the starburst to the blue luminosity. We think that members E and F of the group are candidate tidal dwarf galaxies because of their high metallicity, their kinematics, and the absence of underlying old stellar populations. Finally, we propose that HCG 31 is suffering several almost simultaneous interaction processes. The most relevant of these processes are (1) the merging of members A and C, which would have produced two optical tidal tails, and (2) a fly-by encounter between G and the A+C complex, which would have produced an H I tidal tail from the stripping of the external gas of A+C, from which members F and E have originated. ID="FN1"> 1Based on observations made with several telescopes operated on the islands of La Palma and Tenerife by the Isaac Newton Group of Telescopes, Nordic Optical Telescope and Instituto de Astrofísica de Canarias in the Spanish observatories of Roque de Los Muchachos and Teide of the Instituto de Astrofísica de Canarias.