In the North Volcanic Zone of Iceland, we studied with the greatest possible detail the complete structural architecture and kinematics of the whole Theistareykir Fissure Swarm (ThFS), an N-S-trending, 70-km-long active rift. We made about 7500 measurements along 6124 post-Late Glacial Maximum (LGM) extension fractures and faults, and 685 pre-LGM structures. We have collected the data over the last six years, through extensive field surveys and with the aid of drone mapping with centimetric resolution. In the southern sector of the study area, extension fractures and faults strike mainly N10°-20°, the opening direction is about N110°, and the dilation amount is in the range 0.1-10 m. In the central sector, faults and extension fractures strike mainly N00-10°, the opening direction is N90-100°, and the dilation amount is 0.1-9 m. In the northern sector, extension fractures and faults strike N30-40°, the opening direction is about N125°, and the dilation amount is 0.1-8 m. The variations in strike are attributable to two processes: the interaction with the WNW-ESE-striking Husavik-Flatey transform fault and Grímsey Oblique Rift (Grímsey lineament), and the structural inheritance of older NNE- to NE-striking normal faults. Most extension fractures show a minor strike-slip component: a systematic right-lateral component can be accounted for by the interaction with the WNW-ESE-striking fault zones and the regional, oblique opening of the rift. We regard dyke propagation as a possible cause for the more complex strike-slip components measured at several other fractures. Cumulated dilation and fracture frequency decrease along the rift with distance away from the Theistareykir volcano, situated in the central sector of the ThFS. This is interpreted as a decrease in the number of dykes that are capable of reaching great distances after being injected from the magma chamber.