Does Alcubierre drive allow time travel? In Alcubierre drive, one can travel apparently faster than the speed of light by "compressing" space in front in the direction of travel. So let's say we have a stationary observer at point A, and an object travels from point A to point B and back, departing A at $t_0$ according to the clock of the stationary observer. Is it possible for one to use Alcubierre drive to arrive back at point A before $t_0$ as far as the stationary observer is concerned?
 A: Alcubierre [1] and others have shown how a suitable shaping of space-time curvature can be used to generate a warp bubble, where the contents of the bubble can be transported to arbitrarily large speeds, even exceeding the local speed of light, without suffering inertial effects.  This can be called an inertialess warp drive.
“Abstract: It is shown how, within the framework of general relativity and without the introduction of wormholes, it is possible to modify a spacetime in a way that allows a spaceship to travel with an arbitrarily large speed. By a purely local expansion of spacetime behind the spaceship and an opposite contraction in front of it, motion faster than the speed of light as seen by observers outside the disturbed region is possible. The resulting distortion is reminiscent of the 'warp drive' of science fiction. However, just as happens with wormholes, exotic matter will be needed in order to generate a distortion of spacetime like the one discussed here.” [1]
This well-known paper has inspired other “warp drive” designs [2], however, all of these share a common defect in the requirement for “exotic matter” to provide the proper shaping of the required space-time curvature.  This exotic matter defect is shared with worm hole techniques.
To answer your question: any FTL drive can be used to create closed time-like loops.  With these you can travel back in time, but you get stuck in a loop.
All of this is mathematically consistent with the field equations for General Relativity, but probably not possible in our universe due to the inability to establish the proper conditions.
[1] Alcubierre, M. "The Warp Drive: Hyper-fast Travel within General Relativity,"  Classical and Quantum Gravity, 11(5), L73-77 (1994).
[2] Everett, Allen, and Roman, Thomas, "Time Travel and Warp Drives", University of Chicago Press, Chicago and London (2012).
