Let $k$ be a number field. We provide an asymptotic formula for the number of Galois extensions of $k$ with absolute discriminant bounded by some $X \geq 1$, as $X\to\infty$. We also provide an asymptotic formula for the closely related count of extensions $K/k$ whose normal closure has discriminant bounded by $X$. The key to both of these results is a new upper bound on the number of Galois extensions of $k$ with Galois group $G$ and discriminant bounded by $X$; we show the number of such extensions is $O_{G,[k:\mathbb{Q}],\epsilon} (X^{ \frac{3.5}{\sqrt{|G|}}+\epsilon})$ for any $\epsilon >0$. This improves over earlier work of Ellenberg and Venkatesh, who proved a bound of the form $O_{G,k,\epsilon}(X^{\frac{3}{8} + \epsilon})$.