Number fields are fields which are finite extensions of Q. They come with a canonical invariant called the discriminant, which can be thought of as the volume of a certain canonically associated lattice. While these objects are central to modern number theory, it turns out that counting them is extremely difficult. More precisely, what is the asymptotic behavior of N (n,X) the number of degree n field extensions of Q with discriminant at most X as X grows, while n remains fixed? It is conjectured by Linnik that N (n,x)∼ c _n×, and this has been proven for n<=5 by Davenport-Heilbronn(n=3) and Bhargava (n=4,5) using the theory of pre-homogenous vector spaces. Bhargava has conjectured a precise value for c n. We will explain these developments, as well as recent joint work with Arul Shankar that gives another proof in the case n=3, and which yields a strong heuristic reason to believe Linniks conjecture with Bhargava's value for c n . Along the way, we shall also mention the parallel story in function fields, where much more is known thanks to the theory of Etale cohomology.

This lecture is aimed at a general audience.