It is becoming increasingly common to see large collections of network data objects -- that is, data sets in which a network is viewed as a fundamental unit of observation. As a result, there is a pressing need to develop network-based analogues of even many of the most basic techniques already standard for scalar and vector data. At the same time, principled extensions of familiar techniques to this context are nontrivial, given that networks are inherently non-Euclidean. I will present a number of results extending the notion of asymptotic inference for means to the contexts of various types of networks, i.e., both labeled and unlabeled, and either single- or multi-layer. These results rely on a combination of tools from geometry, probability theory, and statistical shape analysis. I will illustrate drawing from various applications in bioinformatics, computational neuroscience, and social network analysis under privacy.