Anchor apparatus for a flexible and resilient clear orthodontic appliance constructed to be removably fitted over the teeth of a user by friction in a predetermined position. One or more first anchor devices are secured to the bone in the mouth of the user in a position to be adjacent to the orthodontic appliance when fitted over the teeth. The orthodontic appliance is constructed to removably engage the first anchor device or devices when the orthodontic appliance is fitted on the teeth in the predetermined position to removably secure the orthodontic device in the predetermined position. The orthodontic appliance may comprise one or more second anchor devices mounted thereon for removable engagement with the first anchor devices. The first and second anchor devices comprise female recessed portions or male heads that are complementary in shape to facilitate the removable engagement thereof.