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Abstract Given a graphon $$W$$ and a finite simple graph $$H$$ , with vertex set $V(H)$ , 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 Hladký, Pelekis, and Šileikis [17] 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 with scaling $$n^{|V(H)|-\frac{1}{2}}$$ . On the other hand, if $$W$$ is $$H$$ -regular, then the fluctuations are of order $$n^{|V(H)|-1}$$ and the limiting distribution of $$X_n(H, W)$$ can have both Gaussian and non-Gaussian components, where the non-Gaussian component is a (possibly) infinite weighted sum of centred chi-squared random variables with the weights determined by the spectral properties of a graphon derived from $$W$$ . Our proofs use the asymptotic theory of generalised $$U$$ -statistics developed by Janson and Nowicki [22]. We also investigate the structure of $$H$$ -regular graphons for which either the Gaussian or the non-Gaussian component of the limiting distribution (but not both) is degenerate. Interestingly, there are also $$H$$ -regular graphons $$W$$ for which both the Gaussian or the non-Gaussian components are degenerate, that is, $$X_n(H, W)$$ has a degenerate limit even under the scaling $$n^{|V(H)|-1}$$ . We give an example of this degeneracy with $$H=K_{1, 3}$$ (the 3-star) and also establish non-degeneracy in a few examples. This naturally leads to interesting open questions on higher order degeneracies.