It is well known that the class group of a number field is of size bounded above by roughly the square root of its discriminant. But one expects by conjectures of Cohen-Lenstra that the the n-torsion part of this group should be much smaller and there have recently been several papers on this by prominent mathematicians. We present in our talk some of these results and a partial improvement of an estimate of Bhargava, Shankar, Taniguchi, Thorne, Zimmerman and Zhao.