Statistical inference is a mature research area, but distributed inference problems that arise in the context of modern wireless sensor networks (WSNs) have new and unique features that have revitalized research in this area in recent years. The goal of this paper is to introduce the readers to these novel features and to summarize recent research developments in this area. In particular, results on distributed detection, parameter estimation and tracking in WSNs will be discussed, with a special emphasis on solutions to these inference problems that take into account the communication network connecting the sensors and the resource constraints at the sensors.