This thesis studies two types of inhomogeneous random graphs, so-called rank-1 models and the weighted random connection model. Under suitable parameter choices, both random graphs exhibit a scale-free degree distribution as observed in real-world complex networks. For both models, we study the sizes of large components in the subcritical regime. In the rank-1 case, we also establish quantitative Poisson approximation results for cycle counts. The latter allow us to deduce the asymptotic distributions of the lengths of the shortest and of the longest cycle in the subcritical regime.