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We review the formalism of primordial black holes (PBHs) production and showthat the mass variance at horizon crossing has been systematicallyoverestimated in previous studies. We derive the correct expression. Thedifference is maximal at the earliest formation times and still verysignificant for PBH masses $\sim 10^{15}$g, an accurate estimate requiringnumerical calculations. In particular, this would lead to weaker constraints onthe spectral index $n$. We then derive constraints on inflationary models fromthe fact that primordial black holes must not overclose the Universe. This isdone both for the scale-free case of the power spectrum studied earlier and forthe case where a step in the mass variance is superimposed. In the former casewe find various constraints on $n$, depending on the parameters. In the lattercase these limits can be much more strengthened, so that one could find from anobservational limit on $n$ a constraint on the allowed height of the step.Comment: 14 pages, 4 figures; typos corrected, one ref. added. To appear in  PR

Primordial black holes from inflationary models with and without broken scale invariance