Number fields are a central topic of number theory, and yet they are surprisingly difficult to count. We will discuss the history of progress toward counting number fields, and give a new bound on number fields of degree less than 94. The improved bound is achieved through a combination of harmonic analysis and modified sieve methods. We’ll also discuss how similar techniques have been useful in bounding the exceptional set in Hilbert’s irreducibility theorem; that is, in counting the number of irreducible polynomials without a full Galois group.