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Today's update introduced Relics, which are items, of which one can drop per ascension after level 99.

I noticed on the overview page, that there is now also something called "Forge Cores". What are they, how do I get them and what do I do with them?

SCreenshot of the items

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Currently, they do nothing. In the future they will be used to upgrade relics. You get them by salvaging relics.

http://clickerheroes.wikia.com/wiki/Relics

What do Forge Cores do?

As of v0.19, Forge Cores does absolutely nothing. Fragsworth (ClickerHeroes dev) has vaguely mentioned that "forge cores will be used to upgrade items". By items, it's safe to assume he means relics. As far as the scope of what Forge Cores will be capable of, only time will tell when it's implemented. Right now it's just all speculations. How do I get Forge Cores?

You can obtain Forge Cores when you salvage a relic. The amount of Forge Cores a relic is worth depends on the rarity as well as the level of it. However, the value of a relic seems to leans towards the level of it more, rather than the actual rarity. As an example, a level 50 common relic will give more forge cores than a level 40 rare relic.

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According to the game's developer (from Reddit - /u/Fragsworth):

Forge cores don't do anything yet. They'll be used later to upgrade your items.

Quotation from Reddit with more discussion:

They'll supposedly be used to upgrade relics in the near future, but for now they do absolutely nothing.

You get them by salvaging relics in your junk pile (i.e. "deleting" them). I think the rarer the relic, the more forge cores you get.