  We revisit the cosmological and astrophysical constraints on the fraction of
the dark matter in primordial black holes (PBHs) with an extended mass
function. We consider a variety of mass functions, all of which are described
by three parameters: a characteristic mass and width and a dark matter
fraction. Various observations then impose constraints on the dark matter
fraction as a function of the first two parameters. We show how these
constraints relate to those for a monochromatic mass function, demonstrating
that they usually become more stringent in the extended case than the
monochromatic one. Considering only the well-established bounds, and neglecting
the ones that depend on additional astrophysical assumptions, we find that
there are three mass windows, around $4\times 10^{-17}M_\odot,$ $2\times
10^{-14}M_\odot$ and $25-100M_\odot$, where PBHs can constitute all dark
matter. However, if one includes all the bounds, PBHs can only constitute of
order $10\%$ of the dark matter.
