Breakfalling is the part of what I teach that I have hardest time getting compliance from my students in regular practicing, done right it is uncomfortable at times, it is scary at times and general the movements themselves are less intriguing and cool feeling then jumps, swings, ground flow or flips. I am studying some dance approaches to breakfalls to add more of an interesting aesthetic and flow aspect to my breakfall work but fun or not it must be done.
Fear is our greatest limitator, when we teach our body to feel comfortable taking a wide variety of falls we can move with so much greater freedom and joy.
Once a base of breakfalling practice is developed standing grappling is great stimulus for falling ability and high level grapplers have incredible falling skills, though those skills are primarily appropriate.
In general rolls and full body break falling techniques should be taught first, as they have greatest margin of error in providing a wide surface over which to distribute force, however a well developed acrobatic capacity can allow twisting, and flipping to return you to your the feet in situations that would otherwise require a hard breakfall.
These are higher level techniques and should not be the first thing taught but understanding the application of acobratic in more alive movement circumstances is a powerful tool for the high level mover.
Here is the breakfalling practice showing these elements.
Here is some judo applications in live competition
Some of the conditioned responses here are not safe applications outside of this context were landing on the belly is rewarded by the rules while landing on the back or side is not. However the overall ability to deal with inversion and flip or twist to attain the feet that is shown here is extremely valuable.
If your new to breakfalls start here
If you have a well developed acrobatic and interactive practice then definitely start playing with some of this stuff.