Inflation and Primordial Black Holes

We review conceptual aspects of inflationary scenarios able to produce primordial black holes, by amplifying the size of curvature fluctuations to the level required for triggering black hole formation. We identify general mechanisms to do so, both for single and multiple field inflation. In single field inflation, the spectrum of curvature fluctuations is enhanced by pronounced gradients of background quantities controlling the cosmological dynamics, which can induce brief phases of non--slow-roll inflationary evolution. In multiple field inflation, the amplification occurs through appropriate couplings with additional sectors, characterized by tachyonic instabilities that enhance the size of their fluctuations. As representative examples, we consider axion inflation, and two-field models of inflation with rapid turns in field space. We develop our discussion in a pedagogical manner, by including some of the most relevant calculations, and by guiding the reader through the existing theoretical literature, emphasizing general themes common to several models.