Flying Slam is Bowser's Side Special Move in Super Smash Bros. Brawl, replacing the Koopa Klaw. Flying Slam is essentially the same move as Koopa Klaw, with one fundamental difference: while Bowser claws at his opponent as before, if he connects, he'll perform a flying suplex, rather than biting his opponent as he did with the Koopa Klaw.
Overview
Bowser's air movement can be controlled greatly during this move. This move can be used to easily cause a Suicide KO (commonly known as "Bowsercide"), even while in the air, which is particularly useful against edgeguarders performing a "Wall of Pain" style edgeguard.
The victim isn't entirely at Bowser's mercy, though, as they can also alter Bowser's horizontal trajectory - whoever has the lower % will have more control over Bowser's movement. On rare occasions, if the Bowser player and another player are falling off stage it may connect, and can be used as a recovery move. Additionally, when he is in the higher control port, Bowser's death after the Bowsercide can be stalled by mashing the jump button while holding the control stick in one side-direction. This will cause him to perform a second jump, not unlike the motion of a Klawhop. This can be extended further by using Whirling Fortress, and can be used to survive a Bowsercide on stages like Castle Siege (when the stage is about to change) and Norfair (when the lava is at its lowest point).
Interestingly, the DOJO!! has stated that if one uses this move to do a Suicide KO while both are on their last stock, the game would be called in Bowser's favor. However, outside of the PAL edition, this only happens when Bowser's controller port is lower (such as Bowser is player 1 and the opponent is player 2). If Bowser's port is higher, Sudden Death occurs. In the PAL version, and by tournament rules, Bowser is always declared the winner.
Since foes in the Subspace Emissary cannot be grabbed by special moves, Bowser does not actually perform a flying slam. He merely inflicts damage with his claws.
Giga Bowser can also use this move in mostly the same way.
Koopa Hopping
Koopa Hopping is a technique in Brawl that allows Bowser to jump continuously from platforms without the delay of landing. This is achieved by jumping during a Flying Slam just before touching the ground. Hopping somewhat increases Bowser's airborne mobility, among other benefits. Using this, one could perform a long string of aerial attacks, or simply negate punishment from a failed Flying Slam. Giga Bowser can also Koopa Hop.
Various other names for the technique include:
"Klaw Canceling" by KevinM on Smash World Forums.
"Claw Hopping" by Kagato on VGDistict.
"Gimpyfishing", after the professional Bowser mainer Gimpyfish.
"The Turtle Hurdle" by StealthKiller and Phate of Smash World Forums.