we will discuss recent developments in the probabilistic and spectral approaches for graph limits. in particular, we will extend the notion of quasi-randomness, which concerns a class of equivalent properties that random graphs satisfy. for example, we will give several necessary and sufficient conditions for a graph to be the union of two or more quasi-random graphs. one of these characterizations involves eigenvalues and scalable eigenspaces, that we call "graphslets", which dictate the behavior of graph limits for both dense and sparse graphs.