Exchangeable random graphs, which include some of the most widely studied network models, have emerged as the mainstay of statistical network analysis in recent years. Graphons, central objects in graph limit theory, provide a natural mechanism for sampling such graphs. Motifs (patterns of subgraphs), such as edges and triangles, and eigenvalue distributions encode key structural information about network geometry. In particular, it is well known that motif counts (network moments) identify a graphon (up to isomorphisms), hence, understanding their sampling distribution in graphon-based random graphs is pivotal for nonparametric network inference. This thesis develops a general framework for asymptotic analysis of network moments in graphon-based random graphs. We introduce a regularity criterion for motifs that dictates whether their limiting distribution is Gaussian or non-Gaussian. Leveraging multiple stochastic integrals, we derive the joint asymptotic distribution of any finite collection of motif counts—covering both nondegenerate (Gaussian) and degenerate (Gaussian and non-Gaussian) cases—and establish structure theorems on degeneracies with connections to extremal combinatorics. Next, we propose a novel multiplier bootstrap for graphons that consistently approximates the limiting distribution of network moments in all regimes. Combined with a degeneracy-testing procedure, this yields joint confidence sets for any finite collection of motif densities, offering a versatile toolkit for inference in graphon models. As an application, we consider the detection of global structure (testing whether a graphon is constant) via small-subgraph statistics. Invoking celebrated results on quasi-random graphs, we construct a consistent test and derive its asymptotic distribution under both the null and alternate hypothesis. Finally, we study the largest eigenvalue of adjacency matrices from Lipschitz graphons and reveal a dichotomy in its asymptotic behavior. If the leading eigenfunction is nonconstant, the suitably normalized largest eigenvalue converges to a Gaussian limit; if constant, it converges to a non-Gaussian law expressible as a weighted sum of independent chi-squared variables plus a Gaussian component. As a consequence, we extend this dichotomy to random kernel matrices, broadening and reinforcing existing results in the literature.