To solve this, science would need to completely understand the nature of gravity. We don't yet.

A French scientist has developed a sort of "anti gravity" device that he calls a "lifter." He has published detailed plans for small lifter-powered craft and theory of operations on his web site. Over the years, he has built larger and more capable aircraft based on his research. And countless home experimenters around the world have successfully duplicated his results. We're still a few decades away from "flying cars," but what he and people around the world have accomplished is amazing! http://jnaudin.free.fr/lifters/main.htm

I suppose magnets do the same thing, they either engadge or force each other away.

Magnets. BIG magnets. Possibly electromagnets that would run along train tracks. They already have the floating high speed trains in Asia that use magnets.

mythbusters couldnt find a machine that would work.

don't know very cool shit though....

If you had a material that was dense enough to be the equal of the Earth's mass then that would possibly counter the effect of gravity in a local space. Or a big enough Tesla coil!

Lots of helium balloons.

you have to anti warp the curveture of spacefabric. or you have to have an object that doesn't care about gravity. all the others just overcome the effects of gravity. 1st) if you had either an area that OWED mass to that area in theory the space fabric would warp the other way. meaning if Mass=Xgravity so 5 mass = 5 gravity and 0 mass =0gravity then -Mass= -gravity. gravity seems to work by warping the spacefabric so at that exact point if you warped it the other way then it would cause a counter bubble. just like your bank account is + with money it is - when you owe money the second is use an object that already warps gravity the opposite way (proverbial dark matter) that repels instead of attracts. this still attracts itself seemingly but repels normal matter. last is the theory of superstrings and in that theory is that everyknown principal of an object is directed in strings, weight mass gravity. so if you could remove the gravity string you would quintencentially make it have no weight or gravity effect but the mass would still be their. this doesn't effect e=Mc^2 because if the amount of known mass is proportional to how many mass strings an object has and the amount of gravity strings it has in all matter then of course the amount of everything would look that way. however we could simply remove the graviton or the gravity string. +2

Dilithium crystals? ;-)

Opposing magnetic feilds

A Complete knowledge of all details, even down to the smallest, of the interactions in our own space-time continuum... which I dearly wish I had! Hope this helps.

There is nothing that can neutralise the effect of gravity, except gravity from the opposite direction. However there are many things that can create lift. The best known of these is the helicopter. There are others, like helium or hot air balloons etc., but all of them work by creating a different force. In short, even in a helicopter or any of the other devices, although you're lifted you're still standing on the floor of the aircraft or balloon's basket. Also, many of the other new technologies consume a huge amount of energy for the amount of lift they generate.

Since gravity exists, you don't overcome it by ignoring it. You overcome it typically by overpowering it. A helicopter in that sense is an anti-gravity device; you use it to overpower the effect of gravity that would otherwise hold you to the ground. If you want a job making this kind of anti-gravity device, apply here: http://www.sikorsky.com/vgn-ext-templating-SIK/v/index.jsp?vgnextoid=5289e39d40a78110VgnVCM1000001382000aRCRD

We need to get out of the physical and learn to bi locate. The ancients know this trick and many on the earth can do this at will. We don't need to carry all this crap around and learn how to float it. Look up Padre Pio and Bilocation. This is an actual account of a modern day saint that could and did bilocate. He was seen in several places at once.

You mean the Japanese railway system?

This would be an excellent idea to apply to automobiles. Less gravity, better gas mileage. Only problem: someone would tamper with it and off into outer space we go. never to return.