Freerunning

Freerunning is a form of acrobatics in which participants use cities and their urban landscape to perform movements through structures. Though particular form of movement was utilized primarily by Assassins, several factions were also known to be adept users of Freerunning such as Agiles, Theives, Papal Guards, Jannisaries Borgia Messengers, and, in some rare cases, Citizens.

High Middle Ages
During the Third Crusade, Altaïr Ibn-La'Ahad was able to scale any wall if its surface was not too flat and possessed anything he could have grabbed onto. In addition, he could have used any of the following movements:
 * Swing was a movement in which Altaïr swung onto a bar and let go, enabling him to grab another object, or to drop to the ground.
 * Drop was the act of dropping from a hanging position or bar.
 * Wallpass was the act of climbing a tall wall by kicking its surface once to propel upwards.
 * Climb-up was the movement of climbing an obstacle from a hanging position so Altaïr's upper body was above the obstacle, supported by his arms.
 * Pass involved quickly and smoothly passing over medium sized obstacles.
 * Breakfall was a forward roll where the hands, arms, and diagonal of the back contact the ground.
 * Arm jump was to land on the side of an obstacle in a hanging position, hands gripping the top edge of the obstacle, holding the body.
 * Long jump was a jump from a high height, and was usually followed by a roll.
 * Gap jump was similar to a long jump, except over an open expanse.
 * Precision jump was a moving jump from one object to a precise spot on another object, usually having a small surface.

Italy
In addition to all of the movements of the Third Crusade, Ezio Auditore learned new moves throughout his life during the Renaissance.

Fast climb
By moving in High Profile, Ezio could climb structures much faster, making him scale buildings faster than Altaïr. However, the scheme of the ledges he grabbed had to be regular, and he must have started in a standing position on a ledge to do so.

Climb Leap
"Begin in a hanging position, then jump to the ledge above you. As you reach it, grab it with your hand and pull yourself up!"

- Rosa

The Climb Leap movement was first shown to Ezio by Rosa in Venice, and allowed him to grab potentially out-of-reach handholds by performing a vertical jump while climbing.

Sprint Jump
The Sprint Jump was a maneuver unlocked in the Bonfire of the Vanities downloadable content, but could only be used in the Oltrarno District of Florence. The Sprint Jump allowed Ezio to jump twice as far as he usually could by sprinting straight onto and leaping from a springboard.

Swinging
Ezio could also swing from lanterns and pots hanging off of structures in order to navigate cities quicker.

Ottoman Empire
In Constantinople, Ezio was able to perform all of the basic functions he could during the Italian Renaissance.

Hookblade
Ezio was able to perform the climb leap in Constantinople, but used the Hookblade to pull himself up rather than relying on his own brute strength. Also instead of swinging around a corner with lanterns or flower pots he had the choice to use it as a trapeze to swing forward.

Colonial America
Due to the presence of forests in America, and the lack of metropolitan cities, the Assassins there had learned to climb trees, and swing using branches.

Modern Times
Desmond Miles and the other modern day Assassins are able to free run, and perform all of the freerun movements from the Third Crusade, as well as the fast climb movement from the Renaissance.

Trivia

 * Altaïr's Codex sketch indicates ancient natal charts inspired free running.
 * There is a bug present in Assassin's Creed, Assassin's Creed: Bloodlines, Assassin's Creed II, Assassin's Creed: Brotherhood and Assassin's Creed: Revelations in which civilians in certain areas and circumstances are able to perform freerunning movements.
 * In Assassin's Creed: Brotherhood, while trying to tackle Lucrezia Borgia, there is occasionally a bug that makes her free-run over the gardens.
 * In Assassin's Creed: Brotherhood, Ezio is able to swing around a wooden beam while free running (if he jumps off of a galloping horse), rather than simply pulling himself up.
 * Despite the movement being named as free-running, it is in fact known as parkour, which is the efficient use of movement to get from one place to another.
 * Strangely, Desmond is unable to perform the Climb Leap maneuver while free-running around Monteriggioni in Assassin's Creed: Brotherhood, despite possessing Ezio's free-running skill in all other respects.