The network arch bridge (NAB) is a new structural form of arch bridge that was devised 60 years ago by the Norwegian engineer Per Tveit, who is now prof. dr. docent emeritus at the University of Agder, Norway. The network arch is a tied-arch (also known as a bowstring-arch) bridge that combines the benefits of tied-arch bridges and trusses in a single system. While in a classical tied arch, the hangers are vertical, in a network arch, the suspension of the deck to the arch is ensured through a network of inclined hangers that intersect each other at least twice. Thus, the core of the NAB is the hanger arrangement that minimizes the bending moment in the arch to very small values, leading to compression in the arch. Compression with only small bending leads to very slender cross-sections for the elements of the bridge, and deep reductions in terms of materials used and economic and environmental costs. This paper reviews the research into the structural form proposed by Per Tveit and extended by researchers and engineers worldwide. The research methodology included bibliometric literature research, obtained by interrogating the ISI Web of Science (WoS) database and the cited references from the articles on WoS. While the first structural form of a network arch is still in use today and it has proven to be a good idea for spans around 100–120 m, engineers worldwide devised new bridge cross-sections. A brief view of the types of bridge cross-section in use today is given, with details about the bridges chosen as representative. Using analysis of Prof. Tveit’s map, Structurae database and literature review, a database of the network arches around the world was created, emphasizing the development of network arches from the perspectives of continental distribution, opening year, number of structures in different structural forms, and bridge purposes. The structural form was assessed from the perspective of materials used for the arch and the tie, span, purpose and number of lanes, the presence/absence of upper wind-bracings and arch disposition in the vertical plane. In the last part of this review, the newest research into the development of the network arch is discussed. In the past 15 years we have seen an acceleration in network arch development from multiple perspectives: new materials used, such as glulam for the arch or carbon fiber-reinforced plastic for the hangers; span lengths of 250 m and 380 m for large bridge widths; architectural constraints that lead to the outward inclination of the arch, that is pleasing to the eye, but difficult to address from an engineering perspective; the most slender arch bridge in the world, with very slender cross-sections for the arch and the tie.