•State-of-the-art review on train running safety on bridges.•International normative criteria regarding the train running safety on bridges.•Review of the main derailment mechanisms and safety criteria.•Effects of natural hazards, crosswinds and earthquakes, in the train stability.•Influence of track defects and induced deformations in the train stability. Train running safety is a major concern among railway engineers, since a derailment may cause significant personal and material damages. This problem becomes more important if the derailment occurs on bridges, especially at high-speeds, where the consequences may be even worse. The sudden development of high-speed (HS) railway networks that occurred at the end of the 20th century and beginning of the 21st century demanded the construction of new lines with large curve radii in order to fulfill the design requirements of this type of transport. By adding this fact to the orography constraints and, in some cases, to constraints related with the lack of construction area and with the high costs of expropriation, several HS lines started to be developed with more than 75% of their length built over viaducts and bridges. Naturally, this relatively new reality led to a significant increase in the probability of a train being exposed to natural hazards that might jeopardize its stability when it is running over an elevated structure. Hence, this paper aims to present a comprehensive literature review of the problematic associated with the train running safety assessment on bridges. The existing normative criteria from different regions of the world related to this topic are summarized in a first stage. Then, the paper gives a brief description of the available train-bridge interaction models needed to explicitly assess the traffic stability, followed by a presentation of the running safety indexes used to assess the derailment risk. Finally, the available applications regarding the traffic stability against different sources of excitation are systematically reviewed and guidance to future research work on this topic is provided.