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Battle on the Broken Isles
Complete 30 Pet Battle World Quests on the Broken Isles.
Battle on the Broken Isles requires you to complete 30 Pet Battle World Quests across the Broken Isles zones, rewarding 10 achievement points and the 
Broken Pet Portal via in-game mail from Thaumaturge Vashreen.
Unlock World Quests at 110
You must be level 110 to access Pet Battle World Quests on the Broken Isles. Only 3 pet battle World Quests are available each day, and the selection is the same across all characters on all accounts — logging onto alts will not give you different quests.
Check Dalaran Daily Fights
One of the three daily pet battle World Quests may appear in Dalaran rather than on the Broken Isles proper. Dalaran Fight Night tamers include Amalia, Bodhi Sunwayver, Heliosus, Splint Jr., Sir Galveston, Stitches Jr. Jr., and Tiffany Nelson, so check Dalaran each day as these count toward the 30 required completions.
Complete 30 of 38 Quests
There are 38 total Pet Battle World Quests in the rotation across zones including Azsuna and Suramar, but you only need to complete 30 of them to earn the achievement. Because the available fights are randomized daily, this achievement is time-gated and will realistically take a couple of weeks to finish.
Note Suramar Quest Interactions
Some pet battle fights in Suramar are tied to quest progress in that zone, which may cause a different or repeated tamer to appear on days when those fights would otherwise be available. Completing Suramar zone quests can resolve this and open up the intended fights.
Tips
15 of the 38 World Quests also count toward the Family Familiar achievement, so completing those first maximizes progress across both achievements simultaneously.
See the video guide for visual walkthroughs of specific tamer fights and recommended pet strategies.