— Location awareness is highly critical for wireless ad-hoc and sensor networks. Many efforts have been made to solve the problem of whether or not a network can be localized. Nevertheless, based on the data collected from a working sensor network, it is observed that the network is NOT always entirely localizable. Theoretical analyses also suggest that, in most cases, it is unlikely that all nodes in a network are localizable, although a (large) portion of the nodes can be uniquely located. Existing studies merely examine whether or not a network is localizable as a whole; yet two fundamental questions remain unaddressed: First, given a network configuration, whether or not a specific node is localizable? Second, how many nodes in a network can be located and which are them? In this study, we analyze the limitation of previous works and propose a novel concept of node localizability. By deriving the necessary and sufficient conditions for node localizability, for the first time, it ...