Most empirical studies of networks assume that the network data we are given represent a complete and accurate picture of the nodes and edges in the system of interest, but in real-world situations this is rarely the case. More often the data only specify the network structure imperfectly -- like data in essentially every other area of empirical science, network data are prone to measurement error and noise. At the same time, the data may be richer than simple network measurements, incorporating multiple measurements, weights, lengths or strengths of edges, node or edge labels, or annotations of various kinds. Here we develop a general method for making estimates of network structure and properties using any form of network data, simple or complex, when the data are unreliable, and give example applications to a selection of social and biological networks.