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The Broken Isles region is made up of five main regions and smaller areas like the new-but-not-really hub city of Dalaran. Here's the full list:
The Broken Shore: The initial invasion point for the Legion, and where the pre-patch event takes place.
Stormheim: Vrykul, Titans and a lot of Wrath of the Lich King-style architecture - plus a showdown with the God King.
Highmountain: A rough land for rough people, like Tauren and other cows. Deathwing used to chill here.
Val'Sharah: Malfurion's home-zone and where you'll get your first glimpses of the Emerald Dream - and Nightmare.
Azsuna: Dead Night Elves, Naga and their queen Azshara await you here.
Suramar: Once a Night Elf city, this is the endgame zone of the expansion and where the story - at least initially - will conclude.
Dalaran: Making the long trip from Northrend, Dalaran is our new capital city.
It's worth noting that there's no definitive order for these zones. Once you've acquired your artifact, you're given the opportunity to choose between Stormheim, Highmountain, Val'Sharah and Azsuna. New technology means it will scale to your level - as will any dungeons there - giving you freedom. Once you've done the quests in all four of those zones, you'll get to go to Suramar. Through that time, you'll also be returning to Dalaran semi-regularly.