All 28 set dungeons with their two objectives and the conditions for basic completion and Mastery, four for every class, taken from the game files of patch 2.8.0.
Complete one objective and kill 515 monsters within the time limit
Mastery
Complete both objectives and kill all monsters within the time limit
One dungeon per class set
The list holds 28 set dungeons, four for each of the seven classes. Every dungeon belongs to exactly one class set and is named after it: the Dungeon of the Legacy of Raekor goes with The Legacy of Raekor, the Dungeon of the Bones of Rathma with Bones of Rathma. Each entry shows both objectives and the conditions for basic completion and Mastery, worded as in the game.
Two objectives of very different kinds
Every dungeon has two objectives, and they vary a lot. Many ask you to use one specific skill in a specific way: in the Legacy of Raekor dungeon you have to hit 15 different enemies with a single Furious Charge 6 times. Other objectives are restrictions. Three dungeons forbid taking a certain type of damage at all: physical damage in Wrath of the Wastes, Fire damage in Uliana's Stratagem and Poison damage in Helltooth Harness. In two dungeons no enemy may get into melee range, namely Embodiment of the Marauder and Zunimassa's Haunt. And in the Raiment of a Thousand Storms dungeon speed is what counts: you have to reach the golden chest at the end in under 2 minutes.
Basic completion and Mastery
Basic completion needs one objective. On top of that you have to kill a set number of monsters within the time limit, and that number varies widely: 55 in the Spirit of Arachyr dungeon, 515 in the one for Vyr's Amazing Arcana. Eight dungeons ask for more than 300 kills. Mastery has the same condition in all 28 dungeons: complete both objectives and kill all monsters within the time limit. The texts don't state how long that limit is, so this page doesn't either.
What to look out for
Read both objectives before you enter a dungeon, and pay special attention to conditions that apply for the whole run. Examples are keeping Sweeping Wind up the entire time (Monkey King's Garb) or never dropping below 50% Discipline (Unhallowed Essence). Objectives like these shape how you play the whole dungeon, not just its final stretch. In some dungeons both objectives share a theme: in Vyr's Amazing Arcana both take place in Archon form, in Grace of Inarius both revolve around Bone Armor, and in Trag'Oul's Avatar both are about life you spend and heal with your abilities. All numbers in the objectives appear exactly like this in the game files.
What is the difference between basic completion and Mastery?
For basic completion you finish one objective and kill a set number of monsters within the time limit. Mastery requires both objectives and killing every monster within the time limit.
How many set dungeons are there in Diablo 3?
The game files of patch 2.8.0 contain 28 set dungeons, four for each of the seven classes. Each one belongs to a class set.
How many monsters do I need to kill for basic completion?
It depends on the dungeon: anywhere from 55 in the Spirit of Arachyr dungeon to 515 in the Vyr's Amazing Arcana dungeon.