Why is the King's Gambit not played often? I have observed that the King's Gambit is usually not played at international level.  Is there any specific reason for this?
One of the more common lines for the King's Gambit is accepting, followed by 3. Nf3.  That is, 1. e4 e5  2. f4 exf4 3. Nf3.  After this, black can follow with 3... d6, 
the so called Fischer defense.  Fischer was extremely confident that this was a bad position for white.  He wrote an entire article about this line, and famously said, "In my opinion the King's Gambit is busted. It loses by force."  On top of this he added, "Of course White can always play differently, in which case he merely loses differently."

I am not sure about other lines, but likely top grandmasters have analyzed them, and come to similar conclusions that white's position is not a good one.

Remark: Interestingly Fischer himself played the gambit a few times with great success, but he opened 3. Bc4.