The first few videos are (vaguely) instructional; scroll down for the more amusing and entertaining highlights videos.
A summary of the key points of racing can be found in the following instructional video, courtesy of Cmdr furrycat.
A more detailed overview can be found in this tutorial video, prepared by Cmdr Alec Turner for the Lavecon Buckyball Race 2017.
The following tutorial on the "Loop and Swoop" gravity braking approach was created by Cmdr Samwell Drakhyr.
For an alternative take on gravity braking, see the following example from Cmdr Ozric.
And if you're wondering how effective these gravity braking approaches really are, Cmdr furrycat put together this demonstration of the speed difference between different supercruise approaches.
Top left = throttle in the blue zone throughout
Top right = throttle at maximum until 6s, then adjusting as appropriate to maintain that ETA (the six second rule)
Bottom left = throttle at maximum, fly by and loop around
Bottom right = throttle at maximum until 5s then loop and pass close to gravity source to slow dramatically (gravity braking; note that this is a fairly extreme demonstration, the videos above are mid way between the "loop" and "gravity braking" examples in this comparison)
Additional guidance on how to fly fast can be found in the post-race discussions in each of our race threads, with some of the key articles collated in the Buckyball Flight Academy.
Cmdr Alot (Esvandiary) presents a compilation of highlights from the Buckyball Run:
Cmdr furrycat demonstrates the extreme approach to fuel scooping during The Kessel Run:
Cmdr Timothy Knight (Rankaze) shared a blooper reel from his attempts at Buckyball Run 7:
And no list of videos about the Buckyball series would be complete without Cmdr Alot's hilarious series of videos about his assaults on the top spot of the Buckyball Run A* / A* Challenge: