Multi-agent network localization has broad applications in many fields, such as intelligent logistics, factory monitoring, and swarms of unmanned systems. However, the nonlinearity of sensor node measurements with respect to node positions poses great challenges for localization. This paper surveys existing results on multi-agent network localization. Firstly, we compare linear and nonlinear localization approaches, as well as centralized and distributed approaches, from which we can see many advantages of distributed linear localization. Then, we review recent progress on localizability conditions and distributed localization algorithms for multi-agent networks with common measurements such as distances, bearings, local relative positions, and interior angles. Multi-agent network localization remains an active area with many open problems that are discussed finally.