Is time travel possible? Is it possible to go back in time? I read somewhere that according to relativity, it is possible - involving black holes and other stuff - to jump into the past. Is it possible for anything to go back in time either continuously or by jumping?
 A: Yes, it is possible in quantum mechanics. Such setups are known as closed timelike curves (CTCs). As you may know, time travel may lead to the Grandfather's paradox.
There are two mathematical ways to resolve the paradox in QM. The corresponding timelike curves are called D-CTCs (named after David Deutsch who proposed them) and P-CTCs (postselected CTCs).
The D-CTC resolves the paradox by postulating that any paradoxical event converges to a fixed point which is usually a mixed quantum state (like in Schroedinger's cat paradox). The disadvantage of such CTC is that the content of the loop cannot be measured in the future: any measurement collapses the wavefunction and destroys the CTC and does not depend on the input due to inevitable convergence to the fixed point. As such, the information cannot be read or written to/from the loop, the loop just exists by itself.
The P-CTC resolves the paradox by postulating that paradoxical outcomes are prohibited and contradicting histories are eliminated. This has been demonstrated in an experiment. 
Such property of P-CTC allows time travel to be utilized for computation, that is, using the same time interval for multiple calculations. Unfortunately, only one result can be read in the future, so such quantum computer would require special algorithms.
Note that the observer himself never can travel to the past because any histories which lead to entropy decrease would be erased from his memory and as such, unobservable. Whatever the path of the observer, he always sees entropy to increase.
The field that studies closed timelike curves is called non-linear quantum mechanics.
The closed timelike curves in General Relativity, if exist, are thought to resolve paradoxes in a similar way, through non-linear quantum mechanics.
A: It is mathematically possible to create some instances in which an object goes back in time relative to some observer. For example, simply going faster than light causes such an effect, but of course, speed of light is the limit for any massive object.
While it is mathematically possible, there are many paradoxes caused by time travel to past, unless you accept existence of some sorts of multiverses, which doesn't really qualify as a time travel to past anyway, rather travel between said multiverses. Mainly, and most importantly, time travel to past violates "causality", one of the main principles the universe is thought to have, in which the cause precedes the result, e.g. your father is born before you are born. Through time travel to past, it is possible for the "result" to eliminate its "cause" before it causes the result, that is, you killing your father before you were conceived. This is not the only paradox caused by time travel to past, however this is one of the main ones.
Also, I think it was Hawking that suggested this, we see no time-travelers around us, pointing out to the fact that humanity will not achieve the technology to go back in time as far as it wants. HOWEVER, it is possible that time travel to past may be discovered, only limited to taking travelers back to the creation of the machine, as some time travel machines work on this principle, namely wormholes.
Needless to say, wormholes require extremely exotic conditions to form and maintain, and therefore are very far from being created deliberately, if they even exist.
A: There are solutions to Einstein's field equations, which have closed timelike curves. For example Godel's solution. Would that constitute time travel, if you can reach the same point on your world line in finite time? One objection may be that such solution do not describe the universe, but examples as the Tipler's cylinder suggest that at least in theory we could modify parts of the universe to get closed timelike curves. Another objection may be that to talk about past and future you need linear order on time moments, which is not the case for closed curves.  
A: Relativity consists of special relativity (SR) and its generalization general relativity (GR), which includes gravity.
In SR, the history of an object is described by its world-line through spacetime. Every event has future and past light cones, and a material object such as a person's body is limited to "traveling" from the past light cone to the future light cone. Therefore you can't have a world-line that forms a closed loop. A world-line such as a material object's that always stays inside the light cone is called timelike.
In GR the situation is considerably modified. GR allows the existence of closed timelike curves (CTCs), which, although timelike everywhere, nevertheless close back on themselves. To conceptualize this, you can imagine drawing an ordinary timelike world-line on a piece of paper, but then wrapping the paper up into a cylinder. As far as we know, our universe doesn't contain CTCs, and there is no natural or artificial process that could create them. If the universe doesn't already contain any CTCs, then it's believed (Hawking 1992) that creating a new CTC from scratch would require exotic forms of matter (essentially, matter with negative mass). Even if one did have access to exotic matter, creating a CTC would require manipulation of matter on a godlike scale.
Supposing that a CTC exists, there are all kinds of other issues that come up involving causality (time-travel paradoxes) and quantum mechanics. A good (but somewhat out of date) popular-level discussion is given in Thorne 1995.
Hawking, S.W., (1992) The chronology protection conjecture. Phys. Rev. D46, 603-611.
Thorne, Black Holes and Time Warps: Einstein's Outrageous Legacy
A: IMO the answer is no. I share the Israel Pérez view  that time is not an entity 'de per se'. It is a mental construct percieved upon the existence of motion.
See number 7 (pages 6/7) A physicist’s view of the universe: a philosophical approach
Understanding what time 'is', and what it is not, is fundamental to my answer.
Mathematically we can do whatever we like as long we dont contradict, but we need a  physical and philosophical insight when we try to understand the universe. This understanding narrows what can happen. 
The whole paper seems to me very interesting at several levels.
I think that he partially shares the view exposed by Espinosa in 'Ethics' about space/matter (available via the Guttenberg Project).
A: Positrons are electrons that are travelling backwards in time, so, yes, it is possible to travel backwards in time.  Just very expensive. I'm not making this up.
