Figure A is on the edge of a builiding. He wants to shoot figure B on the ground who has a barrier in front of him. Can figure A hit him even though the Barrier is there?

I believe you can shoot over a barrier from an elevated position but I might be wrong.It has been known to happen.

Good question. I would think not, since barrier's are considered blocking terrain.

Wouldn't you have to be standing on top of the barrier to use a ranged combat against someone from that angle?

Figures on the edge of eleveated terain ignore hindering terrain and figure bases. Barrier is blocking terrain, so it is not ignored and would block LOF.

I agree with deusceus. Were about to play and the other players want to hear a judge ruling on this. Judges please?

This just came up at a tournament where I played. I'd never considered it before. However, the judge there ruled that I cannot see the figures behind the barrier. Since it specifically says you can only shoot over the square of blocking terrain you occupy, I agree that you cannot see over the barrier to shoot behind it.

Also, you can't stand on a barrier.

Barrier, as Blocking Terrain, will block LOF between elevated and grounded figures, no matter which is attacking.

That is a quote by HeroComplex and confirmed by Chapeau.

Can you fly over a barrier?

Depends...

Outdoors -- then anyone with Flying, L/C, or phasing can move over/thorough a barrier.

Indoors -- only phasing (as the barrier goes all the way to the roof.)