How do planes fly upside down?

Stunt planes often have symmetrical aerofoils which require a positive angle of attack to develop lift (as opposed to conventional ones that can develop lift with 0 AoA). Otherwise, you could again increase the angle of attack with a conventional aerofoil (flying upside down) or it may be true that sometimes the plane is falling developing 'lift' downwards.
 
There are different types of wings and applications to inverted flying.

You can have an asymmetric profile and a symmetric profile. Aerobatic planes typically have a symmetric wing profile and rely almost entirely on angle of attack to fly.

An asymmetrical wing works inverted by changing the angle of attack to present similar physical characteristics as it does when the right way up - i.e. faster airflow over the upward facing surface than the bottom facing surface and creating greater pressure below.

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A symmetrical wing works constantly by altering the angle of attack and creating the same condition.
 
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how do helimachopters fly upside down?

That's a whole other kettle of fish - Helicopters don't actually fly upside down - not full scale ones anyway. If we're talking about the likes of the Lynx - it's not technically upside down flight, it's more a case of momentum and having such a fast roll and pitch rate that it can make it through the manoeuvre before falling out of the sky. When a helicopter loops it is similar to swinging a tennis ball at the end of a string - as long as there is momentum and inertia keeping the heli moving it will stay up - lose that and it will begin to fall.

Model helicopters have been adapted to reverse the pitch of their main rotors to allow for a reversal of thrust - this has not been adapted for full scale helicopters (as far as I am aware).
 
Basically look at the resultant forces and the which way the big engine on the back pushes ;)

KaHn
 
That's a whole other kettle of fish - Helicopters don't actually fly upside down - not full scale ones anyway. If we're talking about the likes of the Lynx - it's not technically upside down flight, it's more a case of momentum and having such a fast roll and pitch rate that it can make it through the manoeuvre before falling out of the sky. When a helicopter loops it is similar to swinging a tennis ball at the end of a string - as long as there is momentum and inertia keeping the heli moving it will stay up - lose that and it will begin to fall.

Plus the rotors of a Lynx are stiff enough to withstand the forces applied when it rolls. With most helicopters the linkage between the blade & the rotor head would collapse.
 
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