A folded-space transporter was a device that moved objects via a dimensional shift. In the Federation, the principles of folded-space transport, also called adaptive transport, were first described in the Elway Theorem, but it was abandoned in the mid-23rd century because of its dangerous effects on humanoid tissue.
The Ansata separatist movement of Rutia IV used a folded-space transporter called an inverter, based on the Elway Theorem. Ansata leader Kyril Finn credited the inverter with reinvigorating his group despite the fatal results of its use: it allowed his forces to move quickly and with complete surprise anywhere they wished, including areas protected by force fields, and to escape without fear of being tracked.
In 2366, the Ansata abducted Doctor Beverly Crusher of the USS Enterprise-D, hoping that she could find a way to reverse their cellular degradation. They also used the inverter to attack the Enterprise as they tried to force the Federation to intervene on Rutia IV. The Enterprise crew was eventually able to pinpoint the power source of the inverter and rescue Crusher. (TNG: "The High Ground")
The Sikarians of the Delta Quadrant employed a type of folded-space transporter called a spatial trajector, which had a range of some 40,000 light years. The crew of the USS Voyager attempted, without success, to negotiate for this technology in 2371. When some crewmembers obtained a trajector matrix through other means, they discovered the trajector was useless without the Sikarian planet's unique geology and was fundamentally incompatible with Federation technology. (VOY: "Prime Factors") The Borg gained access to the Sikarians' folded-space technology after assimilating Sikaris. (PIC: "The Impossible Box")