  Recent years have seen a growing consensus in the philosophical  community that the grandfather paradox and similar logical puzzles do not preclude the possibility of time travel scenarios that utilize  spacetimes containing closed timelike curves. At the same time,  physicists, who for half a century acknowledged that the general  theory of relativity is compatible with such spacetimes, have  intensely studied the question whether the operation of a time  machine would be admissible in the context of the same theory and of  its quantum cousins. A time machine is a device which brings about  closed timelike curves—and thus enables time travel—where none would have existed otherwise. The physics literature contains  various no-go theorems for time machines, i.e., theorems which  purport to establish that, under physically plausible assumptions,  the operation of a time machine is impossible. We conclude that for  the time being there exists no conclusive no-go theorem against time  machines. The character of the material covered in this article makes it inevitable that its content is of a rather technical nature. We  contend, however, that philosophers should nevertheless be interested in this literature for at least two reasons. First, the topic of time machines leads to a number of interesting foundations issues in  classical and quantum theories of gravity; and second, philosophers  can contribute to the topic by clarifying what it means for a device  to count as a time machine, by relating the debate to other concerns  such as Penrose’s cosmic censorship conjecture and the fate of  determinism in general relativity theory, and by eliminating a number of confusions regarding the status of the paradoxes of time travel.  The present article addresses these ambitions in as non-technical a  manner as possible, and the reader is referred to the relevant  physics literature for details.