Lootboxes drop from events such as Ghost Ship and Cube God. Lootboxes rank from Bronze, Silver, and Gold. Lootboxes also drop from Raids as Elite Lootboxes.

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Elite Lootboxes can give you Onrane, Gold, and Sor Fragment Caches as well as Shards which are used to forge legendaries.
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Bronze, Silver, and Gold Lootboxes give you various tiered loot as well as untiered loot. Gold lootboxes also have a chance to give you shards as well but at a very small rate.
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Premium lootboxes contain Premium Effect chips, which activate a premium effect on one of your characters and can only be opened with Kantos.
Lootboxes can be accessed and opened at the nexus using the button shown right below your rank.XSQbiRl.png