One might think that the potential paradoxes, such as the grandfather paradox, make it pointless to even consider the possibility of backward time travel. However, as we'll see, there are two general approaches that could allow the laws of physics to be consistent even if backward time travel is possible. Each of these is illustrated in numerous works of science fiction, but one or the other must turn out to have a basis in the actual laws of physics, if those laws allow one to build a time machine.
The first possibility is that it could be that the laws of physics are such that whenever you go to pull the trigger to kill your grandfather something happens to prevent it -- you slip on a banana peel, for example (we like to call this the 'banana peel mechanism'). This theory is, logically, perfectly consistent. It is somewhat unappealing, however, because it's a little hard to understand how the laws of physics can always arrange to ensure the presence of a suitable banana peel.
The other approach makes use of the idea of parallel worlds. According to this idea, there are two different worlds: in one you are born and enter the time machine, and in the other you emerge from the time machine and kill your grandfather. There is no logical contradiction in the fact that you simultaneously kill and do not kill your grandfather, because the two mutually exclusive events happen in different worlds.
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Many physicists find the ideas involved in either approach to the solution of the paradox problem so distasteful that they believe, or at least certainly hope, that the laws of physics prohibit the construction of time machines.
