We use a complete sample of about 140 000 galaxies from the Sloan Digital Sky Survey (SDSS) to study the size distribution of galaxies and its dependence on their luminosity, stellar mass and morphological type. The large SDSS data base provides statistics of unprecedented accuracy. For each type of galaxy, the size distribution at given luminosity (or stellar mass) is well described by a log-normal function, characterized by its median  and dispersion σln R. For late-type galaxies, there is a characteristic luminosity at Mr,0∼−20.5 (assuming h= 0.7) corresponding to a stellar mass M0∼ 1010.6 M⊙. Galaxies more massive than M0 have  and σln R∼ 0.3, while less massive galaxies have  and σln R∼ 0.5. For early-type galaxies, the  relation is significantly steeper, , but the σln R–M relation is similar to that of bright late-type galaxies. Faint red galaxies have sizes quite independent of their luminosities. We use simple theoretical models to interpret these results. The observed  relation for late-type galaxies can be explained if the fraction of baryons that form stars is as predicted by the standard feedback model. Fitting the observed σln R–M relation requires in addition that the bulge/disc mass ratio be larger in haloes of lower angular momentum and that the bulge material transfers part of its angular momentum to the disc. This can be achieved if bulge formation occurs so as to maintain a marginally stable disc. For early-type galaxies, the observed σln R–M relation is inconsistent with formation through single major mergers of present-day discs. It is consistent with formation through repeated mergers, if the progenitors have properties similar to those of faint ellipticals or Lyman break galaxies and merge from relatively strongly bound orbits.