To upgrade Equipment rarity, you must use Dust. The Dust cost changes as you upgrade, higher rarity = higher cost.
When a weapons' rarity is upgraded, the base damage increases!

You can also upgrade items, and armor. Most items and armor are upgraded by fusing three of the same rarity together to get a higher rarity.
Armor upgrades with Gem Fragments (📙Upgrading) until it hits Unobtanium, at which point it also requires Dust.

Rarity of WeaponsCost of Upgrade
Common to Uncommon12 Dust
Uncommon to Rare24 Dust
Rare to Super Rare48 Dust
Super Rare to Epic96 Dust
Epic to Legendary192 Dust
Legendary to Mythic384 Dust
Mythic to Unobtainium768 Dust
Unobtainium to Ascended1536 Dust
Ascended to Abyssal3072 Dust
Abyssal to Void1 Void Stone