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After an introduction to motivate the study of primordial black holes (mainly as a dark matter candidate) I will review the current bounds on their abundance. I will then focus on inflationary mechanisms that have been put forward to produce them. I will review the most studied mechanism in single-field inflation, highlighting its shortcomings, and discuss some alternatives. I will also mention some difficulties that currently hamper the obtention of accurate predictions for the primmordial black hole abundance in these models.