Warp drive was a form of faster-than-light travel. Human ships used warp drive by the 26th century and even into the 51st century. The Tenth Doctor implied that the TARDIS made use of one. (TV: Fear Her)
There were at least three types of warp drive: "continuum warp", "implicate theory" and "supralight speed with dumpers". Toroidal time dilators are components of a warp drive. (TV: Warriors' Gate)
Before human life — and all life on Earth for that matter — even began, the Jagaroth created spaceships that used warp drive. The drive could be set at warp 1, warp 2, or warp 3. However, warp 3 would have been suicide if the ship did not have a fully working warp drive. (TV: City of Death)
Human ships of the 26th century were powered by anti-matter, which may have been required for warp drive. Should the navigational computers be tampered with while a ship was in warp drive, it could disintegrate. Warp drive-powered starships were in use up to and possibly beyond the 51st century. (TV: Earthshock, The Girl in the Fireplace)
Warp drive was installed on the freighter XV773 by the kidnappers of Romulus and Remus Sylvest, since the XV class of freighter was never built for it. (TV: The Twin Dilemma)
The Imperial Flagship likely possessed an extremely fast version of this drive as Porridge stated that it could "warp jump" into orbit within seconds of detecting his identification. Indeed, the Imperial Flagship arrived at Hedgewick's World of Wonders and transmatted the survivors of the battle aboard in a matter of ten or so seconds. (TV: Nightmare in Silver)