Currently bitless bridles are not allowed in dressage competitions and are considered unconventional tack for hunters. You do see them in eventing (for the show jumping and cross country phases), in jumpers and in endurance competitions.
Increasingly I've seen discussion about the merits for changing the rules, especially in dressage, to allow the cross-under style bitless bridle to be permitted as an alternative to a bit.
Why ride bitless? I'm not a natural horsemanship devotee, nor do I have anything against bits. I merely could never find a bit that was comfortable for my Trakehner gelding, who has a low palate and a thick tongue. Although my bit box runneth over, my horse spent a good part of every ride focused on his mouth, not on me.
While I'm not someone who stridently believes that shoes are evil, bits are torture and treed saddles will hurt your horse's back, when I came back from a hack yesterday, I realized that Freedom certainly looked the part. 