A number of scientific endeavors of national and international interest today involve populations with interacting and/or interfering units. In these problems, a collection of partial measurements about patterns of interaction and interference (e.g., social structure and familial relations) is available, in addition to the more traditional measurements about unit-level outcomes and covariates. Formal statistical models for the analysis of this type of data have emerged as a major topic of interest in diverse areas of study. Probability models on networks date back to 1959. Along with empirical studies in social psychology and sociology from the 1960s, these early works generated an active community and a substantial literature in the 1970s. This effort moved into the statistical literature in the late 1970s and 1980s, and the past decade has seen a burgeoning literature in statistical physics and computer science. The growth of the World Wide Web and the emergence of online social networking websites such as Facebook and LinkedIn, and a host of more specialized professional networking communities has intensified interest in the study of networks, structured measurements and interference. In this tutorial, I will review a few ideas and open areas of research that are central to this burgeoning literature, placing emphasis on inference and other core statistical issues. Topics include elements of sampling and inference from non-ignorable (network sampling) designs, parametric and nonparametric modeling, and estimation of latent processes on a network, with hints to the applications to social, biological and information networks that motivate these statistical problems.