To upgrade an equipments rarity, you must use Dust. The Dust cost changes as you upgrade, higher rarity = higher cost. 
When a weapons' rarity is upgraded, the damage increases by a set percentage. 

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

Void Stone

Beginning from Uncommon Hellish, you have the ability to stumble across a fractured piece of the void. 
These Void Shards appeared after the Loot Goblins got ahold of a supply of Void Stones. Kill those bastards and complete 
qualifying to have a chance to obtain these shards. Combining 8 of these shards will grant you a Void Stone. You can use this stone on 
Abyssal weapons/gear to upgrade it to the final tier.  Void Stones can be found starting at Epic Hellish. After further research into the origins of these stones, its
been determined that completing a Hellish raid of -REDACTED- difficulty will grant the owner of the key a Void Shard, and increased chances to those who lived to the end!