We present the analysis of a sample of 20 bright low-redshift quasars (M_B<-23 and z < 0.35) observed spectroscopically with the VLT. The FORS1 spectra, obtained in Multi Object Spectroscopy (MOS) mode, allow to observe simultaneously the quasars and several reference stars used to spatially deconvolve the data. Applying the MCS deconvolution method, we are able to separate the individual spectra of the quasar and of the underlying host galaxy. Contrary to some previous claims, we find that luminous quasars are not exclusively hosted by massive ellipticals. Most quasar host galaxies harbour large amounts of gas, irrespective of their morphological type. Moreover, the stellar content of half of the hosts is a young Sc-like population, associated with a rather low metallicity interstellar medium. A significant fraction of the galaxies contain gas ionized at large distances by the quasar radiation. This large distance ionization is always associated with signs of gravitational interactions (as detected from images or disturbed rotation curves). The spectra of the quasars themselves provide evidence that gravitational interactions bring dust and gas in the immediate surrounding of the super massive black hole, allowing to feed it. The quasar activity might thus be triggered (1) in young gas-rich spiral galaxies by local events and (2) in more evolved galaxies by gravitational interactions or collisions. The latter mechanism gives rises to the most powerful quasars. Finally, we derive mass models for the isolated spiral host galaxies and we show that the most reliable estimators of the systemic redshift in the quasar spectrum are the tips of the Ha and Hb lines.