Physical Abilities.

Archery: Damian Jr asserted that his training with his father who was trained by the  League of Assassins gave him greater mastery of the bow than even Green Arrow.

Artistry: Damian Jr is considered as skilled with a brush as he is with a sword.

Business Management: Trained by his in the way of the League of Assassins in controlling a company with an iron grip. Even as a young boy he knows that in business there are wars and even in war there are assassins. Damian is involved with the Wayne Enterprises' board members. He even intimidated the board members by tracking down financial irregularities within the Wayne Enterprises' accounts.

Master computer Hacking

Deception: According to Batman, Robin was a skilled liar.

Disguise: Damian has disguised himself as an elderly busdriver and a substitute teacher.

Driving: He learned to drive at the age of five.

Escapology

Firearms

Genius Level Intellect: Damian has inherited a genius IQ from his father and Grandfather as well as his mother.

Indomitable Will

Intimidation

Investigation: Damian, having been trained under the "World's Greatest Detective", has been able to deduce people's actions as well as solve mysteries with few amounts of data.

Martial Arts: Damian Jr has stated he knows just as many martial arts as Batman and can use them more effectively. This may be a largely overstated comment, but one should not ask for verification. Damian Jr may not be physically tough, but he knows how to control his weight to his advantage in combat. He has stated that he knows, "one thousand ways to kill a man." Damian's training with Batman and the League of Assassins has made him a highly formidable combatant. He is one of the few people capable of fighting on par with Connor Hawke, who is amongst the world's most skilled martial artists.

Karate

Tae Kwon Do

Ninjitsu

Medical Science

Music playing of nearly all instruments at a high level. 

Occultism: Damian Jr knows at a number of magic spell. His mother was his teacher and she taught him and he was an excellent student that excelled in her teachings. 

Peak Human Condition

Pedagogy: He once taught a geology class for a day at Jon Kent's school.

Stealth: Damian Jr was able to sneak into Gotham City and furthermore into the Batcave virtually undetected until he felt necessary. Of all the 
Robins, Damian Jr may be the one that is the most silent.

Stick Fighting

Swordsmanship: On separate occasions his use with his sword, apparently the same sword used by Ra's al Ghul. Although Damian can use his sword and be unmatched with one if armed correctly.

Throwing: Damian Jr is trained by the League of Assassins in shuriken throwing and further trained by Batman in Batarang throwing.