As kermit911 says, radio controlled helicopters can and do fly upside down. The lift of a helicopter is controlles by adjusting the pitch (angle to the horizontal) of the rotor blades. If you go to extremes, then you can produce lift in the opposite direction and keep the model flying upside down.
Real full size helicopters cannot fly upside down and in almost all circumstances cannot loop. There are two main reasons for this, both to do with the rotor blades.
The main reason is that real size rotor blades cannot be made stiff enough to do inverted flight. If you look at the blades of a helicopter as it hovers, you can see a very definite upward curve towards the tip. If you picture this happening the other way around on an upside down helicopter, you'll see that they would chop the tail off! There is no material that will prevent this. The best for the job is carbon fibre, but that's what they're already made of.
Models use rotors with a symmetrical cross section that will happily produce lift in either direction by just adjusting the pitch. Real helicopters need to be a bit more efficient about things and so are designed to be good at producing lift only in the up direction.