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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".

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.

Feels like not far.

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.

Buckminster Fuller agrees with your last point 😊

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

Yep. This spaceship is just great. Comes with radiation shielding, gravity, water, food, breathable air etc. Just remember to bring a star with you, because the ship requires an external power source to keep life support running.

Is a maglev system or quantum locking not counted as anti-gravity?

No. Both of those are using magnetic fields to counter the force of gravity.

They are as anti-gravity as a wing generating lift as it moves through the air.

Can you provide more information on the "immense tension forces"?
The only thing above 1g(what we have on earth) of force would be the outside of the ring.

The tricky parts to me would be:

1. Keeping the structure rigid to avoid buckling.
2. Does the whole structure rotate? How do you dock things to it?
3. If the entire thing doesn't rotate, how do you ensure a perfect seal over a long period of time so that you're not venting atmosphere?

Yes, there are rotating rooms designed to simulate gravity:
https://www.youtube.com/watch?v=bJ_seXo-Enc

Well, the world has gravity pretty much pat down.

Yeah, we have plenty of materials that can withstand 1g of consistent force, because that's the default on Earth's surface and the conditions that we've most studied for material properties.

How do you dock things to it?

On Babylon 5, the docking was done through an opening in the centre, along the axis of rotation, so ships wishing to dock only needed to match rotation and the station would appear to be, well, stationary.

That said, the defence fighters launched from the exterior walls of the station, so I assume they had to constantly adjust in order to re-dock there, assuming they didn't also return through the mid-port.

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.

Spinning is definitely a close cousin to gravity but objects in a spinning reference frame don't fall straight down. Whereas water poured on earth will fall straight down towards the gravitational center, water in a spinning gravitational reference frame will arc. So, definitely close, but not a one to one approximation.

You just have to make the spinny thing bigger!

Yeah, that was just badly worded. Didn’t really think it through either. Let me clarify.

You don’t want to spin the wheel too fast, because that’s going to make people sick. To get around that problem, you need to compensate by making the wheel bigger. More specifically, it should be hundreds of meters in diameter. That means it’s going to require a lot of construction materials, and that gets really expensive to build and even more expensive to deliver.

Normally, space things are built to be as light as possible. Building stuff out of thin aluminium, magnesium, titanium and carbon fiber just won’t be good enough unless you make those structures thicker. The materials are good enough though. If you could launch stuff up there for free, you might as well use steel cables and they would be strong enough.

Nevertheless, it’s still a pressurised donut, and that puts a lot of stress on all the parts, which means you can’t build it out of paper thin titanium. The rotation is totally fine by comparison, since that doesn’t add too much to the stresses.

Dark Energy has entered the chat

"Gravity is a mysterious phenomenon that we're only beginning to understand and will likely never master, baked into the very fabric of the universe. But we can almost perfectly simulate it by spinning real fast :)" will never not be funny to me.

Woah. I want a real-world DBZ gravity room.

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.