Dragonsteel is an important resource in Godfall. You will use it to upgrade and enchant your weapons, trinkets, rings, and amulets. It is reasonably rare, so you will want to find as much of it as you can.
One of the earliest ways to get Dragonsteel is to finish missions. Various missions on each world have Dragonsteel as a completiton reward, making them very important to complete. Mostly, these will be story missions, but you should always check every mission node on the map to see what rewards it is offering.
The supply of Dragon Steel from this source will obviously be quite limited, and thankfully there is another way to get it. If you visit your Forge in the Sanctum, you will be able to break down weapons by salvaging them. If you hover over a weapon, you can see exactly what kind of resources you can get from breaking them down. Higher tier weapons will reward you with Dragonsteel when they are salvaged.
