Federation transwarp drive
The USS Excelsior was employed by Starfleet to test the Federation's version of the transwarp drive in the 2280s. (Star Trek III: The Search for Spock) By the following decade, however, the ship used a standard warp drive. (Star Trek VI: The Undiscovered Country) Dilithium used in all Federation warp cores becomes unstable at the high warp frequency essential in the Federation transwarp design. Achieving warp 10 was never possible on the Excelsior. (VOY: "Threshold")
Flight of the Cochrane
In 2372, a new form of dilithium was discovered from an asteroid field in the Delta Quadrant by the crew of the USS Voyager, which stayed stable at high frequencies. In transwarp simulations at speeds exceeding warp 9.7, tritanium depolarization created a velocity differential between the nacelles and fuselage of the ship. Lt. Tom Paris of Voyager managed to adapt the quantum warp theory and the multispectral subspace engine design by setting up a depolarization matrix around the fuselage of the craft. In this design, the craft was first taken past warp 8 with traditional warp drive, and then switched over to transwarp drive for the warp 10 acceleration.
The Class 2 shuttle Cochrane was taken on the first manned flight through the warp 10 threshold into subspace. Piloted by Paris, he described the time he spent in infinite velocity as for a moment being everywhere. The crew of Voyager hoped to return home earlier than expected, but navigation inside infinite velocity was still something that wasn't tested. The first test flight returned the Cochrane into the exact same spot in space where it had crossed the barrier.
Effects of crossing the transwarp threshold on a Human soon became apparent as Paris went into cellular mutation. His DNA began to rewrite itself within a day through what appeared to be possible future Human evolutionary stages over the coming millions of years. Even killing Paris with high levels of radiation, while trying to stop the transformations, didn't stop the mutation that quickly revived him to life again. Finally becoming mentally unstable, Paris kidnapped Captain Kathryn Janeway and took her on another transwarp journey; this time, he was able to navigate the ship and emerge near a jungle planet. By the time Voyager found them, they had both mutated into amphibian-type creatures and reproduced. The offspring were left in their new environment.
Highly focused antiproton radiation was used in controlled five-second bursts to deteriorate all mutated DNA in the bodies; thus forcing all mutated cells to revert to the original DNA coding that was still present in small traces in the amphibian bodies of Paris and Janeway. The original genome stabilized three days after treatment. (VOY: "Threshold")
Borg transwarp drive
The Borg had established a whole transwarp conduit network, that allowed any type of craft, or even a simple probe such as a log buoy, to travel at transwarp velocities. Even though such networks existed, Borg starships were also equipped with transwarp drives that generated conduits when pre-existing conduits were not available. (TNG: "Descent", "Descent, Part II")
This type of transwarp drive utilized a combination of the traditional warp drive, navigational deflector and deflector shields as its components. The ship was taken to warp factor 2.3 to generate the initial required large subspace field. The navigational deflector was modified to use a tachyon matrix within it to emit tachyon bursts on a high-energy band. The deflector shield of the ship was used to emit the tachyon bursts into space at the resonant transwarp frequency level. The subspace field of the conduit then formed in front of the ship.
A Federation propulsion system, such as the class 9 warp drive of an Intrepid-class starship, however would experience erratic power fluctuations during the transwarp frequency emission. The fluctuations lead to a core pressure increase. This in turn resulted in an unstoppable flood of tachyons to the core and ultimately a core breach, unless the core was ejected. The Borg used transwarp coils as the power source during the generation of transwarp conduits and did not experience this problem. Any warp-capable Federation ship could also utilize Borg transwarp coils. (VOY: "Day of Honor", "Dark Frontier")
When a ship was generating a conduit it was traveling within transwarp space and could easily detect the residual and active transwarp signatures of other ships that were generating conduits or had generated conduits. The area of normal space within a conduit was called a matter stream. (VOY: "Dark Frontier")
When a huge vessel such as a Borg cube used a conduit, it encountered extreme temporal stresses. To maintain a temporal sync inside the ship, a chroniton field needed to be projected throughout the hull from specially-designed conduits in the vessel. Using a recalibrated warp core to generate the field from chroniton-infused bio-neural circuitry, the same effect could be achieved on a Federation ship. However ships like the USS Voyager and the USS Enterprise-D were small enough to stay in temporal sync without this modification. (VOY: "Shattered", TNG: "Descent") A ship as large as a Borg cube was also subject to extreme gravimetric shear once it entered the conduit it was generating. To compensate, the cube projected a structural integrity field in front of it using navigational deflectors. Smaller vessels did not experience this problem. (VOY: "Inside Man")
Once a conduit had been generated, both of its apertures could be re-opened from the outside with a high-energy tachyon pulse emission on a specific tachyon frequency matrix that alternated between the low and secondary bandwidth frequencies. In this way the conduits could be utilized by any type of craft. A ship as large as a Template:ShipClass starship would experience a slight power drain during conduit travel. When exiting a transwarp conduit the exit aperture opened automatically. (TNG: "Descent")
Voth transwarp drive
The Voth have also perfected their own transwarp drive design. Each starship was equipped with a transwarp drive. This technology produced no harmful effects to DNA or an expansion of awareness. Voth ships could also move between regular warp and transwarp. As a ship exited transwarp, a spatial displacement was registered if the ship was not cloaked. (VOY: "Distant Origin")
|aperture • beaming • chamber • coil • computer • conduit • drive • factor • hub • network • probe • signal • signature • space • theory • threshold|
In the script of Star Trek, the warp drive of the alternate reality USS Enterprise is called trans-warp at one point by Spock. Though the line from the script: "By recommending a full stop in trans-warp in the midst of a rescue mission?" was changed to "By recommending a full stop mid-warp during a rescue mission?" for the final cut of the film. 
Canonically it has never been stated that the Excelsior didn't achieve limited transwarp flight, only that it did not achieve infinite velocity. The Star Trek: The Next Generation Technical Manual (page 14) states that the Transwarp Development Project was simply unable to surpass the "primary warp field efficiency barrier" in the early 2280s. It is not clear what this barrier actually was. According to the Star Trek Chronology (pages 75 and 76), the project was deemed unsuccessful by Starfleet Command to the degree that the Excelsior was refitted with a standard warp drive. An editor's note in the Chronology speculates the line by Data: "There has not been a systems-wide technological failure on a starship in 79 years", in TNG: "Evolution", was a reference to the result of the failure of the Excelsior's transwarp drive in 2287.
According to Mr. Scott's Guide to the Enterprise (page 112), the Federation transwarp drive designed for the Excelsior was based on partially recreating the interphasic rift encountered by the USS Enterprise, in 2268, in the episode "The Tholian Web". During the escape from the Tholian tractor field through the brief drop into the interphase, it was recorded by sensors that time would move over 5 times faster inside the interphase for a ship traveling at warp. After the incident, research began on the possibility of creating artificial interphases for travel purposes. With the use of an intricate warp and transporter field matrix, it was possible to open entry and exit points. As the interphase space is completely devoid of matter and energy the ship can cruise inside it on a direct heading to its destination. While time passes for the crew as it would during normal warp travel, upon emerging from interphase a shorter period of time has passed in normal space. The advantage of this type of transwarp is therefore only relative.
This would not be entirely consistent with the what has been canonically established about travel in spatial interphase or interspace as it was also called. In "The Tholian Web" the Enterprise was however able to cross 2.72 parsecs (almost 9 light years) in an instant making some link with that tear and the Federation transwarp design plausible. Also the fact that the Borg use interspatial manifolds to support the structure of a transwarp hubs seems to suggest a link.
Bridge graphics created by Michael Okuda for the bridge set of the USS Enterprise-A in Star Trek IV: The Voyage Home also suggest that the Enterprise-A had at that time a transwarp drive installed. These graphics include a transwarp subsys screen that was visible on camera; and transwarp geometry and transwarp interphase/field stress screens that weren't. Legible versions of these bridge graphics were published in the Mr. Scott's Guide to the Enterprise (pages 122 and 123).
The Spaceflight Chronology (page 181) introduced a similar concept as "superwarp", where the warp factor formula is raised from a cubic formula to the fourth power, based on mathematics of higher dimensions and data on parallel universes. As the Chronology displays two proposed test ships that bear no resemblance to the Excelsior, it is possible this is a completely unrelated technology.
According to DC Comics Who's Who in Star Trek issue 1, the Federation transwarp drive was initially an invention by Zefram Cochrane. Cochrane's notes on transwarp drive were found on the Gamma Canaris planetoid on a follow-up visit by the USS Enterprise in the 2270s.
According to a short story in The Lives of Dax, the Federation transwarp drive was then jointly developed by Starfleet and the Daystrom Institute. It was first tested by Trill pilot Torias Dax in 2284 on the Shuttlecraft Infinity (Shuttlecraft Infinity at Memory Beta, the wiki for licensed Star Trek works).
According to the Pocket Books novel Battlestations!, instead of the dilithium crystals the Federation transwarp drive engine design utilized solid trilithium that retains integrity for workable periods of time. Created in a process of extreme refinement of standard dilithium, it allowed the use of the advanced compacted flow of energy required by the transwarp drive. The Federation transwarp drive was researched for over eight years by Perren, a specialist in interspace physics. In the mid-23rd century, Professor Ursula Mornay perfected the theory for transwarp, though Perren was responsible for constructing the actual hardware for the engine.