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One that's on the fringe of what you're asking is warp drives. Right now it looks like you need ridiculous amount of energy and matter that may or may not exist... But General Relativity is okay with it on principle at least
Not really. It would require negative mass which as far as we know does not exist. And it would generate so much radiation in front of the warp bubble that it would decimate anything nearby when you stopped. There are tons of other major issues with it but those are just 2 I remember off the top of my head.
Of course, if there weren't any problems people would already be trying to build that shit.
Negative gravitational mass is still a theoretical possibility: nothing's ever proven Einstein's equivalence principle. It could be broken for antimatter for example, which could even conveniently explain why there's so little of it (I remember reading that this hypothesis was investigated not long ago but we can't produce and conserve enough antimatter to reliably test that mg=mi)
The second problem isn't an issue if you use it in the vacuum and start and end your trip with classical propulsion.
In fact, the hardest hurdle I'd read on that subject was that with the most efficient warp metrics currently known, you'd still need something like 10^60J for a small spaceship or something ridiculous like that... Orders of magnitude more energy than the mass of the whole solar system.
Which is why I said it was kind of a fringe answer. The fact that physics don't just flat out say "no" is already kind of amazing, which isn't to say that it's definitely possible.
Putting aside the issue that it requires a negative energy density, there's still the issue that it will necessarily violate causality, which is the reason FTL travel is considered problematic in the first place. Maybe it's ultimately okay, but it may also mean that warp drives are fundamentally impossible.