If you haven't already been given the quest, head to the Serenne Stable and speak to Akrah. He's looking to settle an argument, and you're just the hero for the job.
How to find the Eldin Leviathan
This one's pretty hard to miss. Just head to the Eldin Mountains, north west of Death Mountain.
How to find the Gerudo Leviathan
Make sure you've got some cooling items with you (just drop some meat in the snow if you're stuck here) and head south west of Gerudo Town past Dragon's Exile to find the skeleton. You'll have to brave a sandstorm to find it, but once you're close, your Sheikah Sensor will alert you to the Hawa Koth Shrine inside the skeleton. There's also a Great Fairy Fountain here.
How to find the Hebra Leviathan
East of Hebra North Summit, there's a tiny square pool of water visible on the map, which forms part of the puzzle you'll need to solve to find this one.
Use your Cryonis rune to freeze the water, then head up the hill to find a couple of snowballs. Save your game here, as it is possible to mess this up.
Throw the snowball down the hill and over the frozen water to open the gate. You'll find the skeleton and To Quomo Shrine, which has chest with a Royal Claymore inside.
Now head back to the Serenne Stable and collect a Gold Rupee for your troubles.
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