Rollkur , now officially known as "hyperflexion of the neck," is a highly controversial training technique used by some dressage riders today. It was, and is still, used in other equestrian disciplines, most notably show jumping, before being adopted by some well-known dressage riders. However, it is not an old theory in dressage, either: Francois Baucher trained a similar method, although he did all his severe flexing at the halt, and in addition he connected the posture of the hind legs to the mix.
In dressage training, the rider can choose to make the horse work for periods with its neck lowered and its head behind the vertical, for various beneficial reasons, such as suppling, relaxing, and stretching the horse, yet rollkur takes this flexion to the extreme. In rollkur, the horse is asked to lower his head and round its neck as it works--working "deep"--so that the head is coming inward. In the extreme, the horse's mouth touches the middle of his chest. Rollkur is not a quick movement lasting a few seconds, but is held for a length of time, through work at the walk, trot, and canter, including shortening and extension of the gaits. Rollkur is not just longitudinal flexion (nose to chest or forelegs), but accompanied with repeated bending to the rider's leg.
Some riders who use rollkur accomplish the head position by lowering and fixing the hands until the horse yields its jaw backwards in response to the pressure on the bit. This technique (pulling in) goes against all classical riding. In classical dressage, the horse accepts the bit and the horse decides to come down with its head because it trusts the hands of the rider. In good companionship, it is possible to asks the horse to go a little deeper than the animal would do itself (until, eventually, rollkur is established).
Advocates claim that top riders do not "pull" the horses head in. They note that the horse can move its head up into the classical position at any moment. If any horse will not lift up its head at the moment the riders asks this, the horse is considered to be behind the bit.
There is also a great debate as to whether rollkur constitutes animal abuse, both physically due to the held over-flexed position, and mentally due to forced submission. Given that a similar practice is longstanding and routinely seen with the use of draw reins in schooling horses for events such as western pleasure, where it is close to being a universal practice (though also controversial in some circles), the debate has major ramifications across different disciplines.