(covers information from several alternate timelines)
The Federation developed transwarp beaming by 2387. This transporter technology allowed the instantaneous beaming of living beings between planets and distant star systems, or even onto a starship traveling at high warp. The theory was postulated and perfected by Montgomery Scott.
The complete transwarp beaming technology was introduced to the alternate reality in 2258 by Ambassador Spock. In 2259, John Harrison implemented this technology to a portable transwarp beaming device. (Star Trek; Star Trek Into Darkness)
Species such as the Voth employed ships that were equipped with transwarp drives. (VOY: "Distant Origin") Similarly, the quantum slipstream drive, developed by Species 116, was a method of propulsion not unlike transwarp drive. (VOY: "Hope and Fear") The Federation also attempted to develop a transwarp drive controlled by a transwarp computer in the USS Excelsior in 2285, termed "The Great Experiment". However, the effort was ultimately unsuccessful in achieving infinite velocity (Star Trek III: The Search for Spock). In 2374, the USS Defiant studied a rare subspace compression anomaly in the hopes of collecting data relevant to the development of transwarp corridors. (DS9: "One Little Ship")
In 2372, Tom Paris of the USS Voyager managed to break the transwarp threshold in the shuttlecraft Cochrane after discovering a new form of dilithium. Although this form of travel was found to have severe, unanticipated side effects, Paris was the first person in Starfleet to cross the transwarp threshold. (VOY: "Threshold")
The Borg used transwarp signals for communication and the Borg transwarp drive was based on generating transwarp conduits through transwarp space. (VOY: "Unimatrix Zero") This technology was first encountered by the Federation in the 2350s by the crew of the USS Raven. This encounter remained unreported at the time. (VOY: "Dark Frontier") In 2370, the ship used by the rogue Borg that were under the control of Lore used a conduit during an attack on Federation space. Geordi La Forge of the USS Enterprise-D likened going through a transwarp aperture to "falling into a fast-moving river and getting swept away by the current." (TNG: "Descent") In 2374, Seven of Nine attempted unsuccessfully to modify the systems of the Federation starship USS Voyager to create a transwarp conduit. (VOY: "Day of Honor")
In 2375, USS Voyager used a transwarp coil stolen from a Borg sphere to create a transwarp conduit to traverse 20,000 light years closer to the Alpha Quadrant, thus cutting about fifteen years off of its journey. (VOY: "Dark Frontier")
|aperture • beaming • chamber • coil • computer • conduit • drive • factor • hub • network • probe • signal • signature • space • theory • threshold|
In TOS: "The Changeling", set in 2267, the term "multiwarp speeds" was attributed to the velocity of approximately Warp 15. This was obviously before Warp 10 was a reference to infinite velocity. Warp 15 was at that time beyond the limitations of the conventional warp drive of the Enterprise. Multiwarp may be a synonym, or rather an older term, for transwarp.
In TNG: "Where No One Has Gone Before", The Traveler used the energy of his thoughts to move the USS Enterprise-D through space at a speed that appeared to surpass warp 10. According to Star Trek: The Next Generation Technical Manual (page 55), the ship didn't actually achieve warp 10 or go beyond, but it did travel at the speed of about Warp 9.9999999996. This would be confirmed in VOY: "Threshold", where Tom Paris becomes the first Human to travel at warp 10, eight years after the events of "Where No One Has Gone Before".
According to the Star Trek: Deep Space Nine Technical Manual (page 14), the Cardassian Union predicted a breakthrough in energy-field stability for transwarp computation speeds in computer core technology, during the 2370s.