A message (a.k.a. rumor) was generated by a source node in a connected network, and gets propagated across the network, in a certain probabilistic manner. When observing the infected nodes at some time, can we correctly find out the source node which originates the propagation? Such rumor source detection problem and its extensions are fundamental to understanding statistical signal processing issues over large scale networks, which are certainly pivotal for numerous applications, ranging from epidemics, information forensics and security, to social networks. In this talk, we provide a survey of the existing research on the rumor source detection problem, and then in particular describe some of our recent works. Finally we conclude the talk with open issues and prospects.