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Anti-gravity: impossible based on known physics. Artificial gravity: doable, just make it spin.

Even theoretically, we have no prospects for manipulating gravity. No way to, say, generate a gravitational force without a lot of mass or remove the gravitational influence of a large mass. The closest we can get is tricks like using freefall-like motion to simulate microgravity within Earth's gravitational field. Or (and I don't think this one's been done, but it's theoretically pretty straightforward, if involved engineering-wise) using rotation to mimic the effects of gravity in microgravity. I suppose one could also argue that being in orbit simulates microgravity even though you're well within the gravitational influence of Earth. Oh, I guess I can also mention that people training to be astronauts sometimes train underwater because that simulates some of the conditions of microgravity. But yeah. The closest we come to what you're asking about is "tricks".

There are ways to cheat though. For example, spinning things can produce the sensation of being in gravity. Spinning stations would face immense tension forces, which make it difficult to find suitable materials to build them out of. Also, those materials need to be lifted up to orbit, so heavy materials won’t do.

See also: 2001: A Space Odyssey

Another option would be constant acceleration, like in The Expanse. Definitely not a very viable option right now as modern rocket fuels still suck. It could take a while to fix this one.

If you want to be extra cheesy, you just have to make your space station as massive as the Earth, and you’ll get plenty enough gravity. There’s an even cheesier shortcut. Just start calling earth your new space station.

Well, the world has gravity pretty much pat down.

We’re not even sure we know what gravity is. The best explanation I’ve seen is that it relates to the entropy between two systems. We know how to counteract this by applying energy (every time I stand up I do this, but it’s not exciting), but not turn it off or reverse it.

Feels like not far.

The other two answers are spot on, so all I can add is how annoyed I get when sportscasters says someone is defying gravity. There are not, in fact, the parabolic arcs that skaters and snowboarders travel are defining gravity, not defying it. Hang time is a myth, 32 feet per second squared isn't just a good idea, it's the law.

The only way we know that gravity acts in nature is based on mass. We have no real grasp of the mechanism, only that mass exerts it on other mass and the greater the mass, the greater the exertion. It's likely that gravity acts in a way that we can not perceive (maybe another spacial plane above the 3rd) and so we are unlikely to truly understand it unless we invest significant effort into a lot of types of non-practical physics research before we can get to even begin the practical stuff.

That's unlikely in current economic systems that heavily rely on capitalim since the majority of investment will always be in things that can create short term profit. Minimal funding is able to get diverted to the pursuit of knowledge that won't yield results for decades or possibly a century. So it will take many centuries at the current investment rate unless some accidental discovery is made that advances it significantly.