  Given connected graph $H$ which is not a star, we show that the number of
copies of $H$ in a dense uniformly random regular graph is asymptotically
Gaussian, which was not known even for $H$ being a triangle. This addresses a
question of McKay from the 2010 International Congress of Mathematicians. In
fact, we prove that the behavior of the variance of the number of copies of $H$
depends in a delicate manner on the occurrence and number of cycles of length
$3,4,5$ as well as paths of length $3$ in $H$. More generally, we provide
control of the asymptotic distribution of certain statistics of bounded degree
which are invariant under vertex permutations, including moments of the
spectrum of a random regular graph.
  Our techniques are based on combining complex-analytic methods due to McKay
and Wormald used to enumerate regular graphs with the notion of graph factors
developed by Janson in the context of studying subgraph counts in
$\mathbb{G}(n,p)$.
