  Given a graphon $W$ and a finite simple graph $H$, denote by $X_n(H, W)$ the
number of copies of $H$ in a $W$-random graph on $n$ vertices. The asymptotic
distribution of $X_n(H, W)$ was recently obtained by Hladk\'y, Pelekis, and
\v{S}ileikis (2021) in the case where $H$ is a clique. In this paper, we extend
this result to any fixed graph $H$. Towards this we introduce a notion of
$H$-regularity of graphons and show that if the graphon $W$ is not $H$-regular,
then $X_n(H, W)$ has Gaussian fluctuations. On the other hand, if $W$ is
$H$-regular, the limiting distribution of $X_n(H, W)$ can have a non-Gaussian
component which is a (possibly) infinite weighted sum of centered chi-squared
random variables, where the weights are determined by the spectral properties
of a graphon derived from $W$. Our proofs use the asymptotic theory of
generalized $U$-statistics developed by Janson and Nowicki (1991).
