Title: The number of D_{4}-extensions of Q

Abstract: We will begin with a summary of how Malle's conjecture and Bhargava's heuristics can be used to develop the "Malle--Bhargava heuristics", predicting the asymptotics in families of number fields, ordered by a general class of invariants.
We will then specialize to the case of D_{4}-number fields. Even in this (fairly simple) case, where the fields can be parametrized quite explicitly, the question of determining asymptotics can get quite complicated. We will discuss joint work with Altug, Varma, and Wilson, in which we recover asymptotics when quartic D_{4} fields are ordered by conductor. And we will finally discuss joint work with Varma, in which we recover Malle's conjecture for octic D_{4}-fields.