Elite Dark Mages are monsters found in the Black Knights' Catacombs. The dungeon is only accessible during and after While Guthix Sleeps. He uses magic attacks, so armour with a decent magic defence bonus is helpful. However, Elite Dark Warriors and Elite Dark Rangers are in the same area, and they may very well attack you too. During Nomad's Requiem, a mage, ranger and warrior are sent to kill Nomad, but he defeats all of them with great ease.
Elite dark mages attack with a large variety of combat spells, the most powerful of which is Earth Wave. They also have the ability to use prayer, so it is suggested to bring at least 2 forms of attack style to kill them with. Eventually they will run out of prayer so players can kill them. It will just take a little longer to do so. Also, there is a short break between their attacks when they switch their protection prayers. Wearing full Dagon'hai robes or full Elite black armour will make them and others in the dungeon non-aggressive.
Their prayers block all incoming damage of the style they are praying against, unlike players' prayers which only reduce damage.
Smite does not help drain their prayer, and neither does the ability Smash.
They will switch prayer against a particular combat style after taking a total of 1000 to 1500 life points of damage since the last prayer change, poison excluded.
They run out of prayer eventually and become vulnerable to every attack style.
They will only run out of prayer after they have been hit 20 times.