The strong collinear polarizability of the A-H bond in A-H···B hydrogen bonds is shown to lead to an enhanced σ-hole on the donor hydrogen atom and hence to stronger hydrogen bonding. This effect helps to explain the directionality of hydrogen bonds, the well known cooperative effect in hydrogen bonding, and the occurrence of blue-shifting. The latter results when significant additional electron density is shifted into the A-H bonding region by the polarization effect. The shift in the A-H stretching frequency is shown to depend essentially linearly on the calculated atomic charge on the donor hydrogen for all donors in which A belongs to the same row of the periodic table. A further result of the polarization effect, which is also expected for other σ-hole bonds, is that the strength of the non-covalent interaction depends strongly on external electric fields.